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Application of limit design to high-strength aluminum alloy beams Allen, David Elliott 1960

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APPLICATION HIGH - STRENGTH  OF  LIMIT  ALUMINUM  DESIGN  TO  ALLOY  BEAMS  by  DAVID ELLIOTT ALLEN B.Sc,  Queers University,  1957  A THESIS SUBMITTED IN PARTIAL FULFILMENT  OF  THE REQUIREMENTS FOR THE DEGREE OF Master of A p p l i e d Science  i n t h e Department of CIVIL ENGINEERING We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e required standard  THE UNIVERSITY OF BRITISH COLUMBIA September, I960  In p r e s e n t i n g  this thesis i n p a r t i a l fulfilment of  the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t t h e L i b r a r y s h a l l make it  f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y .  agree t h a t p e r m i s s i o n f o r e x t e n s i v e  I further  copying of t h i s  thesis  f o r s c h o l a r l y purposes may be g r a n t e d by t h e Head o f my Department o r by h i s r e p r e s e n t a t i v e s .  I t i s understood  that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n  Department o f  GVj|  &*A\\r\e<lY\\nC\  The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 5*. Canada.  permission.  ABSTRACT A l t h o u g h i n v e s t i g a t i o n s have shown t h a t t h e t h e o r y o f l i m i t d e s i g n a p p l i e s t o beams and some frames made o f m i l d s t e e l , i t i s n o t c e r t a i n whether i t a p p l i e s in  t h e same way t o t h e l i g h t a l l o y s .  S t e e l frames s a t -  i s f y t h e l i m i t d e s i g n p r e d i c t i o n o f a f a i l u r e mechanism not o n l y because s t e e l i s v e r y d u c t i l e but a l s o because s t e e l e x h i b i t s s t r a i n hardening. high-strength hardening.  L i g h t a l l o y s such as  aluminum a l l o y e x h i b i t v e r y l i t t l e  strain  Two l o a d t e s t s were c a r r i e d out on redund-  ant beams made o f t h e aluminum a l l o y t o see i f t h e mechanism c o n d i t i o n o f l i m i t d e s i g n was reached b e f o r e ure t o o k p l a c e i n t h e beam.  fail-  Measurements o f beam d e f l -  e c t i o n s and moments a r e compared t o t h e d e f l e c t i o n s and moments p r e d i c t e d by t h e t h e o r y o f i n e l a s t i c b e n d i n g . The  t h e o r y o f i n e l a s t i c b e n d i n g i s based on t h e s t r e s s -  s t r a i n diagram and t a k e s account o f s t r a i n h a r d e n i n g and  a failure strain.  Tables o f u n i t f u n c t i o n s  derived  from t h e s t r e s s - s t r a i n diagram o f t h e aluminum a l l o y are p r e s e n t e d f o r use w i t h t h e i n e l a s t i c b e n d i n g  theory.  I n both t e s t s , t h e mechanism c o n d i t i o n o f l i m i t d e s i g n was reached b e f o r e  f a i l u r e took p l a c e .  Shortly  a f t e r t h e mechanism c o n d i t i o n was r e a c h e d , a f r a c t u r e occurred  i n t h e f l a n g e on t h e t e n s i o n s i d e o f t h e beam.  iii  Thus the type o f f a i l u r e i n d i c a t e s that not a l l s t r u c t u r a l c o n f i g u r a t i o n s w i l l achieve the mechanism c o n d i t i o n . Beyond the l i m i t o f e l a s t i c deformation (17 k i p s load) and up t o a l o a d of about  27 k i p s , the beam mom-  ents were s i m i l a r to those p r e d i c t e d by the i n e l a s t i c bending t h e o r y .  From 27 k i p s l o a d t o f a i l u r e a t 32  k i p s , the moments were d i s t r i b u t e d i n t h e beam d i f f e r e n t l y than the p r e d i c t e d moments due t o the presence o f high shear f o r c e .  The l o a d - d e f l e c t i o n curves are the  same as the curves from the theory, although measured d e f l e c t i o n s were always g r e a t e r .  The u l t i m a t e c u r v a t u r e  at the s e c t i o n of f a i l u r e was g r e a t e r than p r e d i c t e d  from  the t h e o r y . There were some shortcomings o f the t e s t s .  The  t e s t s were o r i g i n a l l y s e t up t o be unfavourable towards the l i m i t d e s i g n t h e o r y .  However, s t i f f e n e r s were added  at the p l a s t i c hinge l o c a t i o n s to prevent web f a i l u r e ,  -.  and the presence o f the s t i f f e n e r s was h e l p f u l i n a l l o w i n g r e d i s t r i b u t i o n o f moments to take p l a c e i n much the same way as s t r a i n hardening does i n s t e e l beams.  Also  the presence o f the s t i f f e n e r s and s t i f f e n e r h o l e s made i n t e r p r e t a t i o n with the i n e l a s t i c bending t h e o r y uncertain.  F i n a l l y t h e r e were some e r r o r s i n measuring  the moments by means o f s t r a i n gauges.  iv TABLE OF CONTENTS  Page 1  INTRODUCTION PART I  Test Arrangement and I n e l a s t i c 'Theory  Bending  1.1  Arrangement o f beam t e s t s  1.2  The i n e l a s t i c bending t h e o r y  1.3  T e n s i o n and compression t e s t s t o determine the s t r e s s - s t r a i n r e l a t i o n o f h i g h - s t r e n g t h aluminum a l l o y  1.4  Inelastic-theory unit functions f o r highs t r e n g t h aluminum a l l o y beams.  II. 1  R e s u l t s o f Beam Test 1  II.2  R e s u l t s o f Beam Test 2  II.3  Moments d u r i n g Beam T e s t 2  PART I I I III.  52  Beam Test R e s u l t s  PART I I  T h e o r e t i c a l P r e d i c t i o n s and F u r t h e r Observations . I '. ". ". ~. ". '. ". .  87  B e h a v i o u r o f Beam T e s t 2 up t o f a i l u r e as p r e d i c t e d by t h e i n e l a s t i c bending t h e o r y  1  III. 2  Moments and d e f l e c t i o n s p r e d i c t e d by t h e theory of l i m i t design  III. 3  C o n f i g u r a t i o n o f the t e s t beam a f t e r  CONCLUSIONS AND RECOMMENDATIONS References Nomenclature 25 F i g u r e s 11  8  Tables  failure 112  V  ACKNOWLEDGEMENT  The e x p e r i m e n t a l work was c a r r i e d out w i t h the a i d o f a g r a n t p r o v i d e d by t h e N a t i o n a l Research Council.  T h i s a s s i s t a n c e i s g r a t e f u l l y acknowledged. The a u t h o r wishes t o e x p r e s s h i s g r a t i t u d e t o  Dr. A. H r e n n i k o f f f o r h i s guidance and c r i t i c i s m i n the e x p e r i m e n t a l work and the w r i t i n g o f t h e t h e s i s . The a u t h o r i s a l s o g r a t e f u l t o Mr. T o n i  Katramadakis  who c o l l a b o r a t e d i n t h e e x p e r i m e n t a l work.  INTRODUCTION In past years the design o f s t e e l beams and frames has' been based on the e l a s t i c beam theory and the use o f a l i m i t i n g s t r e s s or d e f l e c t i o n .  More r e c e n t l y a t t e n t i o n  has been placed on the theory o f l i m i t d e s i g n o r p l a s t i c design.  Two advantages are claimed i n f a v o u r o f the l i m i t  design t h e o r y .  One i s t h a t , provided i n s t a b i l i t y i s not  a f a c t o r , l i m i t d e s i g n more a c c u r a t e l y p r e d i c t s the l o a d and  c o n d i t i o n o f deformation  when f a i l u r e takes p l a c e .  The other i s t h a t f o r redundant beams and frames, l i m i t d e s i g n i s s i m p l e r than e l a s t i c d e s i g n .  Many t e s t s have  been c a r r i e d out on m i l d s t e e l beams and s i n g l e s t o r y frames which support  the c l a i m f o r l i m i t design.1  Although the  l i m i t d e s i g n t h e o r y has been -shown t o be a p p l i c a b l e f o r m i l d s t e e l i t i s not c e r t a i n whether t h i s can be claimed f o r other d u c t i l e m a t e r i a l s such as the l i g h t a l l o y s .  It  i s the purpose o f t h i s t h e s i s t o study the a p p l i c a b i l i t y o f the t h e o r y o f l i m i t d e s i g n t o redundant beams made o f h i g h - s t r e n g t h aluminum a l l o y . U s i n g e l a s t i c d e s i g n , a s t r u c t u r e i s designed so t h a t under the most severe combination  o f l o a d s the g r e a t e s t  s t r e s s i n the s t r u c t u r e does not exceed a l i m i t i n g g i v e n i n s p e c i f i c a t i o n s as the a l l o w a b l e s t r e s s .  stress The  allow-  a b l e s t r e s s i s d i r e c t l y r e l a t e d t o the y i e l d s t r e s s o f s t e e l  2  and includes a f a c t o r of s a f e t y f o r unknown c o n t i n g gencies.  Thus i n u s i n g the e l a s t i c beam theory, the  c r i t e r i o n o f design i s considered  as the load at  which y i e l d f i r s t occurs i n any part of the s t r u c t u r e . However, mild s t e e l has a great reserve of strength and d u c t i l i t y beyond the s t r e s s and s t r a i n at f i r s t y i e l d and, t h e r e f o r e the e l a s t i c theory u n d e r e s t i mates the load and deformation of the s t r u c t u r e when f a i l u r e takes place. To take account of the d u c t i l i t y of m a t e r i a l s such as s t e e l , i t i s assumed i n the theory o f l i m i t design that when the y i e l d s t r e s s i s reached a t some point i n the s t r u c t u r e , the m a t e r i a l can undergo u n l i m i t e d deformation under constant failure.  an  s t r e s s without  In a f l e x u r a l s t r u c t u r e ( i . e . a beam or r i g i d  frame) t h i s means that at a s e c t i o n of maximum bending moment, y i e l d i n g f i r s t takes place at the e x t r e m i t i e s of the cross s e c t i o n and,- as the moment increases, y i e l d i n g takes place p r o g r e s s i v e l y towards the c e n t r o i d of the cross s e c t i o n .  E v e n t u a l l y the whole beam s e c t i o n  a t t a i n s a c o n d i t i o n of y i e l d i n g or p l a s t i c s t r e s s and the beam s e c t i o n i s able to deform without l i m i t .  The  moment corresponding t o the l a t t e r c o n d i t i o n i s c a l l e d the p l a s t i c moment and the deformation c o n d i t i o n i s c a l l e d a p l a s t i c hinge.  I n the theory of l i m i t  design  3  i t i s assumed that the moment i s e i t h e r e l a s t i c or p l a s t i c and there i s no p a r t i a l l y p l a s t i c moment as i n d i c a t e d above. I t i s important to note that the theory of l i m i t  design  assumes u n l i m i t i n g s t r a i n without f r a c t u r e which, of course, i s not a t t a i n e d i n e x i s t i n g m a t e r i a l s and a l s o that i t neglects the property e x h i b i t e d by mild s t e e l o f s t r a i n hardening or the increase of strength beyond the y i e l d strength. The advantage of l i m i t design l i e s not so much i n simple determinate s t r u c t u r e s as i n redundant ures.  struct-  In l i m i t design i t i s assumed that the s t r u c t u r e  i s under simple p r o p o r t i o n a l l o a d i n g , that i s , each of the load items on the s t r u c t u r e increases i n the same proportion.  At any stage of the l o a d i n g i n a f l e x u r a l s t r u c -  t u r e , there are c e r t a i n l o c a t i o n s of maximum bending moment.  As the load increases from zero, the greatest e l a s -  t i c moment becomes p l a s t i c and the beam s e c t i o n forms a p l a s t i c hinge.  The formation of a p l a s t i c hinge reduces  the redundancy of the s t r u c t u r e by one degree.  I f the  s t r u c t u r e i s determinate, i t w i l l deform without l i m i t and such a c o n d i t i o n i s termed a mechanism. s t r u c t u r e i s indeterminate,  I f the  the load w i l l increase bey-  ond the load when the f i r s t p l a s t i c hinge forms u n t i l p l a s t i c hinges are formed at other l o c a t i o n s of maximum moment.  When there are enough p l a s t i c hinges, the s t r u c -  4 ture or part of i t w i l l form a mechanism and deform w i t h out l i m i t .  The formation of a mechanism i s considered  as the f a i l u r e c o n d i t i o n . For example, i n a f i x e d ended beam under uniform l o a d , p l a s t i c hinges at the supports and a t mid span produce a f a i l u r e mechanism.  Thus f o r  redundant s t r u c t u r e s , besides the increase i n strength due t o p l a s t i f i c a t i o n at a s e c t i o n , there i s an increase i n strength due t o r e d i s t r i b u t i o n of moments i n the s t r u c t u r e t o form a p l a s t i c hinge mechanism.  The l o c -  a t i o n of the p l a s t i c hinges can u s u a l l y be determined quite e a s i l y and once the l o c a t i o n of the p l a s t i c hinges i s known, the f a i l u r e load can be determined from s t a t i c s . In many cases the l i m i t design theory provides a simpler a n a l y s i s than the e l a s t i c beam theory. As already pointed out, the theory of l i m i t design p r e d i c t s that r e d i s t r i b u t i o n of moments takes place i n a redundant s t r u c t u r e u n t i l a p l a s t i c hinge mechanism i s formed.  For r e d i s t r i b u t i o n of moments to take p l a c e ,  a l a r g e angle change may be required a t one of the p l a s t i c hinges.  A l a r g e angle change over a short length of beam  r e q u i r e s a l a r g e curvature and therefore l a r g e s t r a i n s i n the cross s e c t i o n .  The theory of l i m i t design f o r r e -  dundant frames therefore r e l i e s on i t s assumption of l a r g e s t r a i n s without f a i l u r e .  Because s t e e l f a i l s a f t e r  5 some d e f o r m a t i o n , t h e r e i s no assurance w i t h o u t  further  i n v e s t i g a t i o n t h a t f a i l u r e w i l l not take p l a c e a t one o f the p l a s t i c h i n g e s b e f o r e t h e mechanism c o n d i t i o n i s reached. The r e s u l t s o f many t e s t s o f s t e e l beams and frames show t h a t the l i m i t d e s i g n p r e d i c t i o n o f r e d i s t r i b u t i o n o f moments t a k e s p l a c e f o r p r a c t i c a l  cases.^  There  a r e two r e a s o n s why s t e e l frames s a t i s f y t h e l i m i t d e s i g n p r e d i c t i o n o f t h e f o r m a t i o n o f a mechanism. course, i t s great d u c t i l i t y .  One i s , o f  The o t h e r r e a s o n i s i t s cap-  a c i t y f o r s t r a i n hardening - that i s the increase o f s t r e n g t h beyond t h e y i e l d s t r e n g t h . s e r v e t o show t h e need f o r s t r a i n  A s i m p l e example w i l l  hardening.  The beam o f F i g . 1 c o n s i s t s of two f l a n g e s and a v e r y t h i n web.  •  h i b i t s no s t r a i n  . —  The m a t e r i a l ex-  l  i  m  |  t  d e s i g n  hardening  and f r a c t u r e s a f t e r a cons i d e r a b l e amount o f y i e l d -  Fig. 1  ing.  D u r i n g e l a s t i c deform-  a t i o n t h e moment a t t h e support i s greatest.  When t h e s t r e s s i n t h e f l a n g e a t t h e  support r e a c h e s y i e l d t h e p l a s t i c moment c a p a c i t y i s immedi a t e l y a t t a i n e d s i n c e t h e web c a r r i e s no bending  stress.  6  The t h e o r y o f l i m i t d e s i g n p r e d i c t s t h a t t h e moment a t the c e n t r e w i l l i n c r e a s e u n t i l i t e q u a l s t h e p l a s t i c moment.  An i n c r e a s e o f moment a t t h e c e n t r e i n c r e a s e s t h e  p o s i t i v e a n g l e change between t h e s u p p o r t s because b e t ween t h e s u p p o r t s t h e beam i s s t i l l e n t i r e l y e l a s t i c . The c o n d i t i o n o f r e s t r a i n t r e q u i r e s that a negative angle change take p l a c e t o e q u a l i z e t h e p o s i t i v e a n g l e change and t h i s must t a k e p l a c e i n t h e p l a s t i c hinge a t t h e support.  However, t h e angle change i n t h e h i n g e  takes  p l a c e over an i n f i n i t e s i m a l l e n g t h o f t h e beam a t t h e support because a t any a d j a c e n t s e c t i o n , t h e moment i n t h e beam i s l e s s t h a n t h e p l a s t i c moment and t h e r e f o r e is elastic. ite  Since  a  f i n i t e a n g l e change r e q u i r e s i n f i n -  s t r a i n i n t h e f l a n g e , a t t h e s u p p o r t , t h e beam w i l l  f a i l a t t h e s u p p o r t b e f o r e t h e moment a t t h e c e n t r e can increase at a l l . The example p o i n t s out t h a t a l t h o u g h t h e mate r i a l may e x h i b i t c o n s i d e r a b l e d u c t i l i t y , s t r a i n hardening  i s r e q u i r e d i n o r d e r t h a t t h e mechanism c o n d i t i o n o f  l i m i t design i s r e a l i z e d .  Mild steel exhibits consider-  a b l e s t r a i n h a r d e n i n g b u t d u c t i l e m a t e r i a l s such as some of  t h e l i g h t a l l o y s have v e r y l i t t l e  strain  hardening.  One o f t h e purposes o f t h e i n v e s t i g a t i o n i s t o s e t up  a  7 s t a t i c a l l y i n d e t e r m i n a t e t e s t s t r u c t u r e made o f a d u c t i l e material having very l i t t l e  s t r a i n h a r d e n i n g , and, t o f i n d  i f r e d i s t r i b u t i o n o f moments t a k e s p l a c e as p r e d i c t e d by l i m i t d e s i g n , o r , i f f a i l u r e t a k e s p l a c e b e f o r e t h e mechanism c o n d i t i o n i s reached . If  f a i l u r e takes place before the l i m i t  design  mechanism c o n d i t i o n i s reached as p o i n t e d out above, i t i s d e s i r a b l e t o be a b l e t o p r e d i c t such f a i l u r e .  A more  exact theory than the theory o f l i m i t design i s t h e i n 2  e l a s t i c bending t h e o r y p r e s e n t e d by Dr. A. H r e n n i k o f f . The t h e o r y t a k e s account o f t h e s t r e s s - s t r a i n curve o f the m a t e r i a l up t o and beyond t h e p r o p o r t i o n a l l i m i t and can p r e d i c t t h e l o a d and d e f o r m a t i o n c o n d i t i o n o f a f l e x u r a l s t r u c t u r e when f a i l u r e due t o e x c e s s i v e s t r a i n t a k e s place.  The i n e l a s t i c bending t h e o r y has s i m p l i f y i n g assump  tions.  T h e r e f o r e , a n o t h e r purpose o f t h e r e s e a r c h i s t o  t e s t t h e i n e l a s t i c bending t h e o r y e x p e r i m e n t a l l y e s p e c i a l l y the moments and d e f l e c t i o n s beyond t h e e l a s t i c l i m i t and the s t r a i n o r c u r v a t u r e a t t a i n e d a t t h e s e c t i o n o f f a i l ure.  PART I TEST ARRANGEMENT AND INELASTIC BENDING THEORY (1)  Arrangement o f Beam T e s t s One set  o f t h e purposes o f t h e beam t e s t s was t o  up a s t a t i c a l l y i n d e t e r m i n a t e t e s t s t r u c t u r e made  of a d u c t i l e m a t e r i a l having very l i t t l e  strain  hardening  and t o see i f t h e l i m i t d e s i g n p r e d i c t i o n o f r e d i s t r i b u t i o n o f moments m a t e r i a l i z e s b e f o r e f a i l u r e o c c u r s . m a t e r i a l e x h i b i t i n g very l i t t l e  A  s t r a i n h a r d e n i n g and which  i s used i n e n g i n e e r i n g s t r u c t u r e s i s h i g h s t r e n g t h aluminum a l l o y 65S-T6.  The m e c h a n i c a l  p r o p e r t i e s o f t h i s mat-  e r i a l i n t e n s i o n a r e g i v e n i n Table 1, t a k e n from one o f the s i m p l e t e s t s d e s c r i b e d i n S e c t i o n 1 , ( 3 ) .  For t h i s  i n v e s t i g a t i o n , two l o a d t e s t s t o f a i l u r e were c a r r i e d o u t on I beams made o f h i g h s t r e n g t h aluminum a l l o y .  Another  purpose o f t h e t e s t s was t o check t h e a c c u r a c y o f t h e i n e l a s t i c b e n d i n g t h e o r y beyond t h e l i m i t o f e l a s t i c deforma t i o n and t h e r e f o r e  measurements o f beam d e f l e c t i o n and  s t r a i n were made d u r i n g t h e t e s t s . If  i n the example o f F i g . 1, t h e u n i f o r m l o a d  were r e p l a c e d by a c o n c e n t r a t e d l o a d a t t h e c e n t r e o f t h e beam, then d u r i n g e l a s t i c d e f o r m a t i o n , t h e moment a t t h e 8  9  support would equal the moment at the centre and p l a s t i c hinges would develop simultaneously without r e q u i r i n g any r e d i s t r i b u t i o n of moments. This p o i n t s out that the t e s t must be arranged so that a considerable amount of r e d i s t r i b u t i o n of moments from the e l a s t i c values i s needed t o o b t a i n the mechanism c o n d i t i o n o f l i m i t design. The two-span t e s t arrangement of F i g . 2a was chosen f o r the purpose of the t e s t s .  The t e s t arrangement i s sym-  m e t r i c a l about the centre support and loads were a p p l i e d i n each span 1/4 of the span length from the c e n t r a l support.  For t h i s arrangement, the e l a s t i c beam moments  are shown i n F i g . 2b, where the r a t i o of the moment over the support to the moment under the load i s 2.55. The theory of l i m i t design p r e d i c t s f a i l u r e by the occurrence of p l a s t i c moments under the loads and over the support. Therefore, when the beam i s . l o a d e d beyond the l i m i t of e l a s t i c deformation, r e d i s t r i b u t i o n of moments must occur u n t i l the moments under the loads equal the moment over the support. Beam Dimensions The dimensions o f the t e s t arrangement are given i n F i g . 2a, and the geometric p r o p e r t i e s of the I s e c t i o n are given i n Table 2. The beam span of 90 inches between the centre support and the outer support f i t t e d conveniently i n the t e s t i n g machine.  The beam s e c t i o n - a  10  6-inch  deep by 3-inch wide I s e c t i o n (designated  by Alcan) was  chosen so that the s t r e n g t h  beam d i d not exceed the  c a p a c i t y of the  28008  of the  test  t e s t i n g machine.  L i m i t Design F a i l u r e Load The  v a l u e of the l i m i t design f a i l u r e  load  w i l l be found on the assumption that the beam i s  unaltered  by the presence o f s t i f f e n e r s or spread supports. moments Mp  over the  support and  f a i l u r e c o n d i t i o n and from e q u i l i b r i u m .  under the l o a d s  Plastic  give  the v a l u e of the l o a d can be  the  found  From the v i r t u a l work e q u a t i o n ,  P  P(3/4L6) - M_(30) + M _ u e ) 6  -  28  T  M  (1)  p  TT  With a p l a s t i c s e c t i o n modulus (Zp) 7  i n i and  a y i e l d s t r e s s (<Ty)  of 39.9  moment of the t e s t beam i s ZpO~y the span L = 90 (1),  30»5 k i p s .  kips or 295  2' in.,  the  the  Loading and  With  i n . the f a i l u r e l o a d P i s , from e q u a t i o n The  o u t e r beam r e a c t i o n i s Mp  t o t a l r e a c t i o n at the  k i p s or 52.6  7.38. plastic  in.-kips.  37fe and  of  centre  support i s  4.2  or  kips,  #  2(30.5-4.2)  kips. Support Apparatus The  beams were supported and  loaded i n a T i n i u s  11  Olsen m e c h a n i c a l t e s t i n g machine as shown i n F i g . 3a.  A  h o r i z o n t a l g i r d e r a t t a c h e d by a p i n t o t h e head o f t h e t e s t i n g machine d i s t r i b u t e d t h e l o a d from t h e machine t o the l o a d p o i n t s o f t h e beam.  The l o a d was a p p l i e d t o t h e  t e s t beam by c y l i n d r i c a l r o c k e r arrangements which r o t a t e d about p i n s a t t a c h e d t o t h e g i r d e r as shown i n F i g . 3a. I f the p o i n t o f t h e beam i n c o n t a c t w i t h t h e r o c k e r moved h o r i z o n t a l l y , t h e r o c k e r would r o t a t e about t h e p i n , and the l o c a t i o n and v e r t i c a l d i r e c t i o n o f t h e l o a d would be maintained.  The l o a d i n g p o i n t s p r o v i d e d r e s t r a i n t t o any  l a t e r a l motion such as l a t e r a l b u c k l i n g o f t h e beam f l a n g e . The l o a d was brought onto t h e beam by t h e v e r t i c a l m o t i o n of t h e head o f t h e t e s t i n g machine. The supported  two o u t e r s u p p o r t s were p r o v i d e d w i t h  pins  by r o c k e r arrangements r e s t i n g on t h e base o f  the t e s t i n g machine. ( F i g . 3b). I f t h e beam c o n t a c t moved h o r i z o n t a l l y , t h e p i n c o n t a c t moved w i t h t h e beam, w h i l e the r o c k e r r o t a t e d on i t s c y l i n d r i c a l base.  The a r r a n g e -  ment p r o v i d e d v e r t i c a l p i n support  and prevented h o r i z o n t a l  f o r c e s b e i n g c a r r i e d i n t h e beam.  F o r t h e f i r s t beam t e s t ,  the c e n t r a l support  r e s t e d on a p i n and p r o v i d e d a c y l i n d -  r i c a l s u r f a c e a t t h e beam contact>  ( F i g . 3c). The a r r a n g e -  ment p r o v i d e d p o i n t r e a c t i o n and, because t h e c y l i n d r i c a l s u r f a c e w i t h t h e p i n c r e a t e d a r o c k i n g s u p p o r t , t h e r e was  12  l i t t l e r e s i s t a n c e t o h o r i z o n t a l beam f o r c e .  To p r o v i d e  a h o r i z o n t a l r e s i s t a n c e i n t h e second t e s t , i t , was d e c i d e d to  make t h e c e n t r a l support a s i m p l e b l o c k support  resting  on t h e base o f t h e t e s t i n g machine. ( F i g . 3a)», A c a l c u l a t i o n which f o l l o w s shows t h a t t h e r e s i s t a n c e o f t h e beam web and s t i f f e n e r was n o t s u f f i c i e n t t o w i t h s t a n d t h e r e a c t i o n expected at t h e c e n t r a l support. second  Therefore, i n the  t e s t t h e c y l i n d r i c a l s u r f a c e was r e p l a c e d w i t h a  f l a t s u r f a c e two i n c h e s wide. Beam S t i f f e n e r s From t h e l i m i t d e s i g n c a l c u l a t i o n , t h e f a i l u r e l o a d expected a t t h e l o a d p o i n t s i s 3 0 . 5 k i p s and t h e r e a c t i o n expected a t t h e c e n t r a l support i s 52.6 k i p s . Assuming f o r t h e p r e s e n t t h a t t h e web o f the beam does n o t b u c k l e , t h e c a p a c i t y o f the beam web w i t h o u t to  stiffeners  a c o n c e n t r a t e d l o a d can be c a l c u l a t e d from t h e web  c r i p p l i n g f o r m u l a i n t h e A.I.S.G. S p e c i f i c a t i o n s .  This  f o r m u l a g i v e s t h e c a p a c i t y as t(N+-2k)0f » 12.5 k i p s  }  where 2  Of  i s t h e f a i l u r e s t r e s s i n compression  (39.9 k i p s / i n . f o r  aluminum a l l o y ) , t i s t h e t h i c k n e s s o f t h e web ( 0 . 2 5 i n . ) , N t h e l e n g t h o f b e a r i n g ' ( N - 0 ) , and k t h e d i s t a n c e from the o u t e r f a c e o f the f l a n g e t o t h e web o f t h e t o e o f the f i l l e t  (0.627 i n . ) .  Thus w i t h o u t c o n s i d e r i n g web  b u c k l i n g , s t i f f e n e r s were needed i n t h e web t o d i s t r i b u t e  13  the force and prevent the web from buckling.  The s t i f -  feners were cut from a T s e c t i o n of the aluminum a l l o y and bolted t o both sides of the web over the centre support and under the load as shown i n F i g . 3 c .  For the  f i r s t beam t e s t a l l of the b o l t s were 1/2 inch diameter. The  c a r r y i n g capacity of each b o l t i s governed by bear-  ing on the web of the beam and the b o l t capacity i s therefore ( 0 . 2 5 ) x ( 0 . 5 ) * 6 4 kips or 8 k i p s , using a bearing f a i l ure s t r e s s of 64 k i p s / i n . f o r the a l l o y .  The t o t a l cap-  a c i t y of the beam and s t i f f e n e r t o a concentrated load i s (12.5 + 6(8))  kips or 6 0 . 5 k i p s which i s s u f f i c i e n t t o  withstand the r e a c t i o n expected at the c e n t r a l support. Unfortunately, the presence o f the s t i f f e n e r s and b o l t holes gave r i s e to d i f f i c u l t i e s i n the a n a l y s i s and i n t e r p r e t a t i o n of the beam t e s t s .  One d i f f i c u l t y  was that at a p l a s t i c hinge l o c a t i o n , the s t i f f e n e r c a r r i e d by way o f the b o l t s an unknown part of the moment i n the beam.  To prevent excessive moment being c a r r i e d by the  s t i f f e n e r i n the second beam t e s t , the four b o l t holes i n the s t i f f e n e r s near the top and bottom were o v a l l e d to allow f r e e h o r i z o n t a l b o l t movement.  In F i g . 4 , when  the beam i s deformed, the two upper b o l t s move outward, and the two lower b o l t s move inward.  The o v a l l i n g of the  holes weakens the bearing capacity of the b o l t on the  14  stiffeners. ^  bolts.  t  Therefore, i n  t h e second beam t e s t , 5/3 i n c h d i a m e t e r b o l t s were used for  t h e f o u r upper and l o w e r  To determine t h e b e a r i n g  capacity o f the b o l t s  on t h e s t i f f e n e r , a l o a d t e s t was made o f a 5/3 i n c h b o l t c o m p r e s s i n g on t h e f l a t load-indentation  edge o f t h e s t i f f e n e r .  The  r e s u l t s are given i n F i g . 5 f o r a s i n g l e  b o l t on two s t i f f e n e r s u r f a c e s , i . e . , t h e same c o n d i t i o n as a s t i f f e n e r b o l t i n t h e t e s t beam.  The c a p a c i t y o f  the b o l t a g a i n s t t h e two s t i f f e n e r s w i l l be assumed a s 6 k i p s - t h e l i m i t o f e l a s t i c d e f o r m a t i o n i n F i g . 5. strength  The  o f t h e f o u r upper and l o w e r b o l t s i s reduced  from 3 k i p s i n t h e f i r s t beam t e s t t o 6 k i p s , and t h e r e f o r e t h e t o t a l r e s i s t a n c e o f t h e beam and s t i f f e n e r t o a c o n c e n t r a t e d l o a d i s [60.5-2(4)] k i p s o r 52.5 k i p s . The c e n t r a l support r e a c t i o n from t h e f a i l u r e l o a d  cal-  c u l a t i o n i s 52.6 k i p s , and s i n c e no a l l o w a n c e i s made for  s t r a i n h a r d e n i n g t h e r e a c t i o n e x p e c t e d w i l l be g r e a t e r .  T h e r e f o r e , i n t h e second beam t e s t , t h e c e n t r a l s u p p o r t was widened t o 2 i n c h e s , g i v i n g t h e beam an a d d i t i o n a l r e s i s t a n c e i n t h e web o f 0 . 2 5 ( 2 ) 4 0 = 20 k i p s , and a t o t a l r e s i s t a n c e o f 72.5 k i p s t o a concentrated load.  C o l l a r s t o P r e v e n t Beam I n s t a b i l i t y D u r i n g t h e f i r s t beam t e s t , t h e upper f l a n g e between the o u t e r s u p p o r t and t h e l o a d p o i n t l a t e r a l l y before reached.  t h e f a i l u r e c o n d i t i o n i n b e n d i n g was  To p r e v e n t f l a n g e b u c k l i n g i n t h e second t e s t ,  c o l l a r s were a t t a c h e d ine.  t o t h e base o f t h e t e s t i n g mach-  The c o l l a r s were made o f a b o l t e d framework o f  a n g l e i r o n s as shown i n F i g . 3a. o f t h e framework p r o v i d e d ion  buckled  Two v e r t i c a l members  r e s t r a i n t t o l a t e r a l beam mot-  and freedom t o v e r t i c a l m o t i o n .  Beam Measurements S t r a i n gauges were a t t a c h e d  t o the flanges of  the beam f o r measuring bending s t r a i n s .  S t r a i n gauges  were e l e c t r i c a l r e s i s t a n c e type w i t h an 1/8 i n c h gauge length  ( P h i l l i p s PR9214).  a l l s t r a i n gauges.  F i g . 6 shows t h e l o c a t i o n o f  Gauges were p l a c e d on t h e f l a n g e s  under t h e l o a d p o i n t s and over t h e c e n t r a l s u p p o r t . The f i r s t t e s t beam had a d d i t i o n a l f l a n g e gauges a t a beam s e c t i o n close to the c e n t r a l support. beam i s s t a t i c a l l y i n d e t e r m i n a t e ,  Since the t e s t  t h e moments i n t h e  beam cannot be found d i r e c t l y from t h e l o a d w i t h o u t t h e use o f a d d i t i o n a l measurements.  I n t h e f i r s t beam t e s t ,  a l l o f t h e f l a n g e gauges were i n l o c a t i o n s o f l a r g e i n e l a s t i c s t r a i n s and t h e y c o u l d n o t be used t o f i n d t h e  16 moments i n the beam.  T h e r e f o r e , f o r the second  test  beam, gauges were p l a c e d on o p p o s i t e f l a n g e s o f the beam a t s e c t i o n s half-way between each support and each load point.  Since o n l y e l a s t i c bending c o u l d o c c u r a t  these s e c t i o n s , the beam moments c o u l d be determined from the f l a n g e s t r a i n s by means o f t h e e l a s t i c theory. be  beam  The extent o f moment r e d i s t r i b u t i o n would then  determined. From the c a l c u l a t i o n o f the l i m i t d e s i g n  ure l o a d , the beam shear f o r c e  fail-  (V) between t h e l o a d  point  and the c e n t r a l support i s 2 6 . 3 k i p s , and the average shear s t r e s s i n the web o f the beam i s c l o s e t o V / A o r w  13.5  k i p s / i n . , where  i s the web a r e a .  At the h o l e s  the value o f V / A i s g r e a t e r than 1 3 . 5 k i p s / i n ? w  yield  I f the  s t r e s s i n shear i s taken as l/^TJ of the y i e l d  stress  i n t e n s i o n , o r 2 3 . 0 k i p s / i n ? , then the shear s t r e s s e s i n the web approach  yield.  From e l a s t i c beam t h e o r y , the  shear s t r e s s a c r o s s the web i s n e a r l y u n i f o r m .  However,  t h i s may not be expected where the beam i s bent  inelastic-  ally,  and l o c a l  than y i e l d .  shear s t r e s s e s i n the web may be g r e a t e r  Shear  s t r a i n s were measured by p l a c i n g a  t e n s i o n gauge and a compression  gauge at 4 5 degrees t o the  beam .axis on o p p o s i t e s i d e s o f the web.  17  One o f t h e purposes o f t h e beam t e s t s was t o check t h e c o r r e c t n e s s o f t h e d e f l e c t i o n s the i n e l a s t i c b e n d i n g t h e o r y .  p r e d i c t e d by  D i a l gauges r e s t i n g on  the base o f t h e t e s t i n g machine measured beam d e f l e c t i o n under t h e l o a d p o i n t s and a t a d i s t a n c e o f 0.45 o f t h e span from t h e c e n t r e s u p p o r t . ment o f t h e d i a l gauges.  F i g . 6 shows t h e a r r a n g e -  18 Table  1.  M e c h a n i c a l P r o p e r t i e s o f Aluminum 65S-T6 i n Tension  (from t e s t shown i n F i g . 9a) kips i n f  Modulus o f E l a s t i c i t y -  9,540  Proportional Limit  36.3  "  Yield  39.9  n  43.4  "  Stress  (0.2  per cent o f f set)  Ultimate  Stress  0.0038  S t r a i n at Proportional Limit S t r a i n a t Ultimate Table  2.  0.088  Stress  S e c t i o n P r o p e r t i e s o f 6 i n . I-beam  (28008 Alcan)  From measurements o f the beam s e c t i o n d - 6.00 i n .  I  b = 3.00 i n . t  = 0.250 i n .  w  T  t f » 0.314 i n . fillet A  radius  5/l6 i n .  =  — X  3.311 i n !  =  The f o l l o w i n g p r o p e r t i e s are needed for  calculation: h = d-tf A^  t^h  a  . r  5.686  At  in.  1.421 i n .  2 2  A  f  1/2 Qi-Aj]  =  -  0.945 i n .  =  x-x Z  .  j2 I a  x-x dA  19.05 i n .  3 -  6o35 i n .  =  7.38 i n ?  19  Loading  and  f>oppo^"^>  2 2 ''z 2Z 2 l/  90"  90"  2V (fcA M o m e n t ^  Undcv  £Ua<$>+ic  Defo^in-ortori  , £.55 M L  /  ML  (O Moment  Undey  ML  the-? Condition  of Limi'f D^igio  Theory  Ca)  £><z<3nn  Test |.  (fc^ Bcdrn Test £.  33V  ®  I©  CD  1  i  23  ( 2 )  The I n e l a s t i c Bending Theory A basic assumption i n the theory of l i m i t design f o r f l e x u r a l s t r u c t u r e s i s that no f a i l u r e takes place before the moments i n the s t r u c t u r e have r e d i s t r i buted from the e l a s t i c moments to form a p l a s t i c hinge mechanism.  I f f a i l u r e does occur a t one of the p l a s t i c  hinges before the mechanism c o n d i t i o n i s reached, i t i s d e s i r a b l e to be able t o p r e d i c t such f a i l u r e .  A more  exact theory than the theory of l i m i t design i s the theory of i n e l a s t i c bending presented by Dr. H r e n n i k o f f .  The  theory i s based on s i m p l i f y i n g assumptions s i m i l a r t o the e l a s t i c beam theory. (1)  They are:  The d i s t r i b u t i o n o f s t r a i n s over the cross  s e c t i o n o f the f l e x u r a l member i s l i n e a r . (2)  The s t r e s s - s t r a i n r e l a t i o n i n bending i s  the same as i n simple tension or compression, and the s t r e s s s t r a i n curves i n tension and compression are i d e n t i c a l . (3)  The f l e x u r a l members are symmetrical about  t h e i r n e u t r a l axes. (4)  The bending moment diagram o f the loaded  s t r u c t u r e i s bounded by s t r a i g h t  lines.  (5)  Normal f o r c e s are ignored.  (6)  Deformations produced by shearing forces  are disregarded.  24  (7)  Instability  is  (S)  Deformations  ignored.  of  the  structure  are  assumed  small.  The upon which in  the  ship  stress-strain  the  theory  form of  is  a graph,  determined  pression  owing t h i s  ections  and  shear  curve.  These  Dr.  find  Hrennikoff  of  the  centres  beam i s of  the  concentrated  and the  to  extremities  following relations  and  flange  The  is  are  the  for  flange  for  =  1  fJ-£^£  where  + K<T  m is  the  defl-  I beams  by  here. the  web  between  the  areas are  of  foll-  change,  I shapes,  the  a  assumptions  web. the  assumed  Therefore, flange  stress.  m  com-  stress-strain  distance  i n terms  and  determined  angle  listed  of  relation-  aluminum a l l o y ,  of  the  (2) cr= 0~CO  as  tension  derived  analysis  flanges  the  Moment  been  equal  the  of  tests  i n terms  assumed  at  equation.  tests  o n l y be  To s i m p l i f y t h e  expressed  curvature,  have  and w i l l  i n assumption  <r*<r&\ t h e a b o v e  moment,  relations  2  or  strength  Having  stresses  used  may b e  from simple  section. to  table,  For high  curve  c a n be u s e d  based  from simple  samples.  stress-strain  is  curve  (2) moment  about  the  strain  25  centroid e  of  the  dimensionless  a n d <r a r e  the  flange  and  K is  web  area.  Angle  the  ratio  A  s t r a i n and s t r e s s -  f  For a given  Change  <^>  A  s  w  I/2I section  the  size  flange  of  ^ % dm  I  «  A  V  point  flange for  of  angle  stant  4> i s  contraflexure  s t r a i n , V the change.  and  shear  the  t  2  h  e  rnAw h  (3)  no  w  angle  a point  force,  (See  Fig.  2 x  np  0  h  (4)  of  change  mQ m o m e n t  a n d nQ t h e  3).  between and  unit  60  function  F o r a member u n d e r  con-  Deflection  £ -  XQ i s  h ^A^  2  where contraflexure €Q flange  deflection.  from the  strain,  (See  Fig.  (5)  mo  where  and  =  l /  moment,  4> =  of  M  to  V  where a  7,  respectively,  area  beam,  in Fig.  Jme  S  the  dm  is  tangent  3).  of  the  h ^A^UQ  =  the  and UQ i s  length  (6)  deflection to  the  a point unit  beam.  at of  a  mQ m o m e n t  function  F o r a member u n d e r  point  for  constant  moment.  Equations for  the  simple  cantilever  (4) a n d  case,  (6) g i v e  deformations  and d e f o r m a t i o n s  for  other  26  bending moment conditions can be derived from combinations of the c a n t i l e v e r . They are given i n reference Shear Stress  T  (2).  - Y. where t i s the shear s t r e s s at a d i s t -  ance ^ from the c e n t r o i d . <^(£) and  are u n i t f u n c t -  ions defined as f o l l o w s : ^co =  «r  = jrjlrde.  4-x<r •  (9)  For shear stresses at the c e n t r o i d ,  The bending f u n c t i o n m, the angle change n^, the d e f l e c t i o n f u n c t i o n U Q and the shear f u n c t i o n s  <^  and  rel-  j^?> , are dependent only on the s t r e s s - s t r a i n  ation  (T(£) and the shape of the I beam as given by K.  Therefore, f o r a given K value, once these f u n c t i o n s are tabulated f o r each value of € corresponding t o the s t r a i n i n the f l a n g e , the moments deformations and shear s t r e s s e s can be found f o r any p a r t i c u l a r case.  The subscript - o i  n  nQ and U Q w i l l be dropped h e r e a f t e r . The beam t e s t s w i l l be used to check the c o r r e c t ness of the. theory of i n e l a s t i c bending with regard t o the moments, d e f l e c t i o n s and f a i l u r e c o n d i t i o n which the theory predicts.  2 7  FI6.7.  FIG. 8.  unit  fo^cfioo  f°K  unit  -function  -for  ^n^lc change de-f lection  23  (3) Tension and Compression T e s t s to Determine the S t r e s s - S t r a i n R e l a t i o n of High Strength Aluminum A l l o y Since no complete s t r e s s - s t r a i n r e l a t i o n r e a d i l y a v a i l a b l e f o r h i g h s t r e n g t h aluminum a l l o y t e n s i o n and compression  men  (65S-T6),  t e s t s were c a r r i e d out on ,a number  of round samples cut from extruded 1 alloy.  was  Three t e n s i o n specimens and  beams made o f the one  compression  speci-  were cut from both a 6 in..X beam (designated 23003  Alcan) and  a 4 i n c h H beam (designated 29001 A l c a n ) . Tension specimens of diameters  i n . , and 0.50  0.24  i n . , 0.30  i n . were cut and machined from the  f l a n g e , and web-flange i n t e r s e c t i o n r e s p e c t i v e l y , loaded to f a i l u r e  i n a h y d r a u l i c t e s t i n g machine.  short e x t e n s i o n s , e l o n g a t i o n was Extensometer.  "web, and For  measured by a Cambridge  The Cambridge Extensometer i s a  mechanical  d e v i c e w i t h a 4 i n c h gauge l e n g t h which e l i m i n a t e s moment e f f e c t s by a v e r a g i n g e l o n g a t i o n on o p p o s i t e s i d e s o f the specimen. two  One  o f the 1/2  i n addition  d i a m e t r i c a l l y opposite e l e c t r i c a l r e s i s t a n c e s t r a i n  gauges with a 1/4 SR-4  i n c h specimens had  i n c h gauge l e n g t h (SR-4  s t r a i n r e a d i n g s were averaged  effects.  type A 7 ) .  The  to e l i m i n a t e moment  For l a r g e e x t e n s i o n s , c a l i p e r s measured elong-  a t i o n over a 4 i n c h gauge l e n g t h .  29  Compression specimens were cut from the webflange i n t e r s e c t i o n of the H and I beams and machined i n t o rods 1/2 inch i n diameter 1 l / 2 inches l o n g .  The s p e c i -  mens were compressed between the f l a t heads of the h y d r a u l i c t e s t i n g machine u n t i l they were h i g h l y deformed. For small deformations, shortening was measured by 2 - l / 4 inch SR-4 s t r a i n gauges placed d i a m e t r i c a l l y opposite. For large deformations, a d i a l gauge r e s t i n g on the lower head o f the t e s t i n g machine, measured the movement of the upper head i n compressing the sample.  The d i a l gauges  were u n s a t i s f a c t o r y f o r small deformations because the surfaces between the sample and the t e s t i n g machine heads f i t t e d unevenly. By t a k i n g a number o f samples, v a r i a t i o n s were taken i n t o account due t o the s i z e of the specimen and due t o v a r i a t i o n o f m a t e r i a l between beams as w e l l as between d i f f e r e n t l o c a t i o n s of the same beam.  An a d d i t i o n a l  cause f o r v a r i a t i o n during i n e l a s t i c deformation i s the e f f e c t of time.  Therefore, the specimens were loaded over  d i f f e r e n t durations of time varying from 20 minutes t o 3 hours. The s t r e s s - s t r a i n r e s u l t s are given i n F i g . 9 f o r the tension t e s t s and F i g . 10 f o r the compression tests.  A  summary of the main mechanical p r o p e r t i e s of  30  each  test  is  given  The The  tension  0.35  per  general  specimens  cent  by a g r a d u a l lastic  behaviour  stress,  of  inelastic  increased  inelastic  passed  at  43 k i p s / i n  about  stress  stress  was  r a p i d l y at  until  then  the  ultimate  Necking took  2  reached  specimen f r a c t u r e d  compression  tests  strain  curves  were  similar  After  some  y i e l d i n g however,  tension  to  increased  inelastic  whereas  decrease stress out,  i n the  slope  continued  to  and a f t e r  of  part  the  curves the  the  stress  was  after  considerable  of  the  of  stress.  stress  the  stress  sectional  in tension  deformation,  f o r m a b u c k l i n g shape  first  the  increase  considerable  was  the  tension in  com-  in  the comAfter  specimen  accompanied  the  began  by  curve.  specimen  deformation,  -  decreased.  stress-strain as  area  compression  and t h i s the  it  ine-  with  rapidly with s t r a i n than i n  t e s t s m o s t l y because  pression some  more  to  place  a lower nominal  In the  increased  the  at  The  place  and a f t e r  followed  40 k i p s / i n ?  about took  follows.  about  deformation.  extension  of  pression  as  33 k i p s / i n .  increase  tests.  is  s t r a i n and about  Large  necking the  tests  up t o  increase  ultimate  the  elastically  little  the  of  extended  deformation  stress.  3.  i n Table  a  The  flattened test  was  stopped.  Since  there  was no s h a r p  yield  point,  the  stress  31  most s u i t a b l e f o r l i m i t d e s i g n c a l c u l a t i o n s i s some a r b i t r a r y y i e l d s t r e s s such as the s t r e s s a t 0.2 set from the i n i t i a l e l a s t i c l i n e . be r e f e r r e d t o as the y i e l d s t r e s s F i g . 9,  per  This w i l l <Ty and,  hereafter  as seen i n  i t c o r r e s p o n d s to the r a p i d change i n the  s t r a i n curve.  The  tension  t e s t s show v e r y l i t t l e  h a r d e n i n g or s t r e s s i n c r e a s e whereas the  stressstrain  beyond the y i e l d s t r e s s ,  compression t e s t s show more s t r a i n h a r d e n i n g .  The  t e s t r e s u l t s g i v e n i n F i g . 9 and  Table 3 show c o n s i d e r a b l e v a r i a t i o n i n the  stress.  Beyond the u l t i m a t e  10  and  physical  e r t i e s such as the modulus o f e l a s t i c i t y and at u l t i m a t e  cent o f f -  the  prop-  strain  s t r e s s where  n e c k i n g t a k e s p l a c e , the v a r i a t i o n o f s t r e s s - s t r a i n among the samples was i s affected  considerable,  m o s t l y because t h e  strain  by the r a t i o o f c r o s s - s e c t i o n a l a r e a to  gauge l e n g t h .  T h e r e f o r e , past the u l t i m a t e . s t r e s s ,  the  s t r e s s - s t r a i n i s shown as a dashed l i n e . Two  tension  t e s t s - Nos.  c a r r i e d out o v e r about 2| hours and out  over 20  - 40 m i n u t e s .  s t r a i n r e s u l t s are due  The  3 and  the r e s t were c a r r i e d  v a r i a t i o n s i n the  to d i f f e r e n c e  6, were  of m a t e r i a l  stress and  size  o f specimen as w e l l as the time a l l o w e d t o l e t c r e e p t a k e  32  p l a c e and t h e r e f o r e v a r i a t i o n s due t o time e f f e c t be determined w i t h c e r t a i n t y .  cannot  On t h e o t h e r hand, by  comparing t h e two 2\ hour t e s t s ' w i t h t h e f o u r s h o r t - d u r a t i o n t e s t s , v a r i a t i o n s due t o time e f f e c t a r e n o t l a r g e w i t h i n t h e l i m i t s 20 minutes t o 3 h o u r s . Stress-Strain Relation f o r Inelastic  Theory  The i n e l a s t i c t h e o r y i s based on a s t r e s s s t r a i n curve which i s t h e same i n t e n s i o n as i n compression.  I n the t e n s i o n s t r e s s - s t r a i n curves, f a i l u r e  o c c u r s a f t e r some d e f o r m a t i o n ; whereas in' t h e compression curves,' s t r e s s i n c r e a s e s w i t h d e f o r m a t i o n w i t h o u t f a i l u r e . T h e r e f o r e , f a i l u r e i n bending w i l l t a k e p l a c e i n t h e t e n s i o n s i d e o f t h e beam and f o r t h i s r e a s o n a t e n s i o n s t r e s s - s t r a i n curve was chosen f o r t h e t h e o r y . The  s t r e s s - s t r a i n c u r v e s from t h e t e n s i o n  t e s t s shov; a r e a s o n a b l y c l o s e agreement b e f o r e t h e u l t i mate s t r e s s and s t r a i n a r e r e a c h e d .  The s t r a i n a t u l t i -  mate s t r e s s v a r i e s w i d e l y from 5 l t o 9 p e r cent and bey- • ond t h e u l t i m a t e s t r a i n t h e r e i s a wide v a r i a t i o n as previously discussed.  For the i n e l a s t i c theory, the s t r e s s -  s t r a i n curve from T e n s i o n Test No. 1 u s i n g t h e SR-4 s t r a i n measurements was chosen a l t h o u g h t h e r e i s no r e a s o n f o r c h o o s i n g one t e s t over a n o t h e r .  33  Modulus o f E l a s t i c i t y The v a l u e s o f t h e modulus o f e l a s t i c i t y from the t e s t s v a r i e d c o n s i d e r a b l y r a n g i n g from 9300 . 2 > t o 11,000 k i p s / i n . A l s o , i n Tension Test Mo. 1,  2 kips/in. there . 2  was a d i s c r e p e n c y between t h e modulus 9540 k i p s / i n .  given  by t h e SR-4 gauges and t h e modulus 9300 k i p s / i n . g i v e n by The modulus 9540 k i p s / i n . i n  the Cambridge Extensometer.  T e n s i o n Test Mo. 1 was chosen as p a r t o f t h e s t r e s s s t r a i n curve f o r t h e i n e l a s t i c t h e o r y . I n Beam Test 2, t h e moments i n t h e beam were determined  by t h e r e l a t i o n M e E Z where Z i s t h e s e c t i o n =  modulus, £ t h e f l a n g e s t r a i n measured by P h i l l i p s and E t h e modulus o f e l a s t i c i t y .  1  gauges  When t h e v a l u e 9540  k i p s / i n ? was used f o r E, t h e moments'determined i n the. beam d i d not agree s t a t i c a l l y w i t h t h e l o a d .  The problem  o f d e t e r m i n i n g t h e moments i s d i s c u s s e d i n d e t a i l i n S e c t i o n I I , (3). To recheck t h e modulus o f e l a s t i c i t y , a 1 2 - i n c h l e n g t h o f t h e I beam was l o a d e d i n a x i a l compression to the e l a s t i c l i m i t .  up  SR-4 gauges w i t h a l / 4 i n c h gauge ;  l e n g t h were p l a c e d s y m m e t r i c a l l y o p p o s i t e a t t h e c e n t r e o f the f l a n g e s and a t t h e c e n t r e o f t h e web.  M i r r o r exten-  someters w i t h a 6-inch gauge l e n g t h were p l a c e d symmetrica l l y opposite a t the centre of the f l a n g e s .  Symmetri-  34  c a l l y o p p o s i t e gauges were averaged.  The s t r e s s - s t r a i n  r e s u l t s g i v e n i n F i g . 11 show agreement between t h e f l a n g e SR-4 gauges and t h e m i r r o r e x t e n s o m e t e r s .  Both s e t s o f  gauges measured a modulus o f e l a s t i c i t y o f 10,200 k i p s / i n . In t h e beam t e s t s , d e f l e c t i o n s were measured at four l o c a t i o n s .  The d e f l e c t i o n s which were measured  d u r i n g e l a s t i c d e f o r m a t i o n can be used f o r d e t e r m i n i n g the modulus o f e l a s t i c i t y and these computations a r e g i v e n i n S e c t i o n I I , (3).  The modulus computed from Beam 2 Test No. 1 was 9600 k i p s / i n . and t h e modulus computed from 2  Beam T e s t No. 2 was 10,100 k i p s / i n .  S i n c e t h e l a t t e r mod-  u l u s was determined d i r e c t l y from t h e second t e s t beam, i t was used \vith t h e f l a n g e s t r a i n measurements f o r computi n g moments d u r i n g t h e t e s t . The. Aluminum Company o f Canada i n a l e t t e r s t a t e d t h a t t h e v a l u e o f £ never v a r i e s a p p r e c i a b l y from 10,000 k i p s / i n .  The v a l u e s measured above however, i n d i -  c a t e t h a t t h e modulus E f o r a sample specimen i s q u i t e v a r i a b l e and depends on i t s l o c a t i o n i n the beam.  35 TABLE 3. MECHANICAL PROPERTIES FROM TENSION AND COMPRESSION TESTS  Specimen Location  Specimen Diameter in.  Duration of test  Modulus o f Elasticity Kip/in?  F i l l e t of I beam F i l l e t of H beam  0.4996  1/2 h r .  0.4997  2/3 h r .  9540 (SR-4) J6.3 9300 (Cambridge) 9550 (Cambridge) 34.5  4.  Flange o f Bbeam  0.2998  1/2 h r .  9300  6.  Web o f H beam  0.2386  2 2/3 h r .  : ,5.  Web o f I beam  0.2394  Flange o f X beam  0.3002  Test No.  Proportion^jfc Y i e l d Stress Ultimate Limit " a t 2$ o f f set ' s t r e s s . Kip/inf .4, Kip/in? Kip/in? 0  Ultimate Strain Per Cent  Failure. Stress Kip/in?  Elongation Per Cent  Area Reduction Per Cent.  Tension 1. 2.  3.  44  39.9.  43.4  9.0  30.4  14.8  64  39.9  42.7  6.8  31.0  11.3  64  (Cambridge) 30.0  38.9  42.0  6.0  25.8  9.5  71  9300 (Cambridge) 31.5  40.0  42.4  5.0  26.0  7.5  73  1/3 h r .  9300  (Cambridge) 33.5  39.5  42.7  6.8  28.7  9.0  71  2 1/4 h r .  9600  (Cambridge) 31.0  39.5  43.0  7.3  27.0  10.5  32.0  39.2  -  -  -  -  -  33.0  40.5  -  -  -  -  -  Compressic »n 1.  F i l l e t of H beam  0.500  2.  F i l l e t of I beam  0.500  Length o f 1 beam  61  9960 (SR-4) 1/2 h r .  11000  (SR-4)  10,200 (SR-4) )- Flange, 10,200(Mirror ex- ) tensometers ) 10,500(SR-4) - Web.  -  ZS - -:4 •  Test Or). :  6f  1 I -:. !  i i 1  I'M'' "'0, i";  I  ' I  5 £ - 4 1 ;gcrd3£3f5 • •  L L• H  -  \  l.i-  -!--,-. -i -a  •iT:u.  I  -ft  1 , . .  11. I-  ' ' t i . . L i L  i-  1 i r: 1 1  iCt> !  i b p • I !  1  i  i '  1  !  ;  i i :  Iii a< ® : : i i i  ! t <  ;jih 1 1 L 11  i ! t  tbli  ' LL ; !  : . . .  'j P :  •: i r  I-i r-i ' i i ;  :  1  p pi  L[Vi  I  !  i  i i  ! :  ; SiJwaiVi I I  a ifor  t i  11  i i  i i  I  41  (4)  I n e l a s t i c - T h e o r y U n i t F u n c t i o n s F o r High Aluminum A l l o y Beams  Strength  W i t h T e n s i o n Test No. 1 chosen f o r t h e s t r e s s s t r a i n curve, the u n i t f u n c t i o n s of the i n e l a s t i c  theory  d e s c r i b e d i n S e c t i o n 1,(2) can be found f o r aluminum , a l l o y 1 beams.  The u n i t f u n c t i o n s m, % n u. and 7  are  7  d e r i v e d from t h e s t r e s s - s t r a i n curve by e q u a t i o n s (6) and ( 9 ) .  (2), (4),  Up t o t h e e l a s t i c l i m i t t h e u n i t f u n c t i o n s  are o b t a i n e d from a s i m p l e r e l a t i o n t o t h e modulus o f e l a s t i c i t y E.  (See r e f e r e n c e ( 2 ) ) . I n t h e i n e l a s t i c r e g i o n  o f t h e s t r e s s - s t r a i n c u r v e , a s t e p by s t e p i n t e g r a t i o n o f s m a l l i n c r e m e n t s must be made.  A set of unit functions  must be computed f o r each shape o f I beam as g i v e n by the parameter K - t h e r a t i o o f t h e f l a n g e a r e a t o oneh a l f t h e web a r e a . The v a l u e s o f K chosen f o r d e t e r m i n a t i o n o f t h e u n i t f u n c t i o n s a r e 0 , 1/2, 1,  1 1/2, 2, 2 1/2, and 3 .  These v a l u e s i n c l u d e t h e r e c t a n g u l a r s e c t i o n (K  0) and  =  the shapes o f I and H beams n o r m a l l y l i s t e d i n t h e manuf a c t u r e r s ' catalogues.  The u n i t f u n c t i o n s f o r K  the v a l u e f o r t h e t e s t beams determined also included.  :  1.330  i n Table 2, a r e  The u n i t f u n c t i o n s m,r\, U and ^ a r e  42  l i s t e d i n T a b l e 4 f o r f l a n g e s t r a i n s above and i n c l u d ing the proportional l i m i t .  The u n i t f u n c t i o n s below  the p r o p o r t i o n a l l i m i t can be computed from t h e r e l a t i o n s given i n reference ( 2 ) .  The s t e p by s t e p i n t e g r a t i o n was  done on t h e Alwac I l l - e computer a t t h e U n i v e r s i t y o f B r i t i s h Columbia.  Curves f o r m,n„ u. and ^it> a r e g i v e n dm  i n F i g . 12 f o r K = 0, 1, 2, 3. The q u e s t i o n a r i s e s as t o what v a l u e o f t h e f l a n g e s t r a i n e. does a beam f a i l i n bending.  When a  s t a t i c a l l y d e t e r m i n a t e beam i s l o a d e d t o . f a i l u r e , t h e f a i l u r e l o a d i s determined moment.  by t h e v a l u e o f t h e maximum  Therefore, since the u n i t f u n c t i o n m i s pro-  p o r t i o n a l t o t h e beam moment, t h e f l a n g e s t r a i n a t f a i l ure corresponds t o t h e maximum v a l u e o f m.  This value of  the f l a n g e s t r a i n i s e q u a l t o o r somewhat g r e a t e r t h a n t h e s t r a i n a t u l t i m a t e s t r e s s depending on t h e v a l u e o f K. I n a s t a t i c a l l y i n d e t e r m i n a t e frame such as t h e t e s t beam o f F i g . 2 a , suppose t h e v a l u e o f m a t the c e n t r a l has passed over i t s maximum v a l u e .  support  Although the f l a n g e  s t r a i n has i n c r e a s e d d i r e c t l y over t h e s u p p o r t , t h e f l a n g e s t r a i n s a t a d j a c e n t s e c t i o n s o f t h e beam have  decreased  because t h e moment o r m a t t h e s e s e c t i o n s d i d not r e a c h the maximum v a l u e .  A l s o , due t o t h e decrease o f moment a t  43  the c e n t r a l s u p p o r t , t h e montent under t h e l o a d must have i n c r e a s e d t o m a i n t a i n e q u i l i b r i u m w i t h t h e l o a d . T h i s means t h a t t h e beam d e f l e c t i o n a t , say t h e l o a d p o i n t has i n c r e a s e d .  B u t , a l l t h e a n g l e change a l o n g  the beam from t h e c e n t r a l support t o t h e l o a d p o i n t which c o n t r i b u t e s t o t h i s d e f l e c t i o n i n c r e a s e t a k e s p l a c e a t o n l y one l o c a t i o n - t h e p o i n t over t h e c e n t r a l support.  S i n c e a n g l e change cannot be accumulated a t a  p o i n t , f a i l u r e i n bending w i l l take p l a c e by f r a c t u r e o f the t e n s i o n -flange. a g a i n determined  T h e r e f o r e , t h e beam f a i l u r e l o a d i s  by t h e f l a n g e s t r a i n c o r r e s p o n d i n g t o  the maximum v a l u e o f m. The u n i t f u n c t i o n s have been computed up t o the v a l u e o f t h e f l a n g e s t r a i n c o r r e s p o n d i n g t o t h e m a x i mum v a l u e o f m, but t h e v a l u e s beyond have n o t been t a b u l a t e d because t h e y do n o t a p p l y c o r r e c t l y . u  and  dm  V a l u e s o f n>  a r e i n c o r r e c t i n these r e g i o n s because s t r e s s  r e c e s s i o n has t a k e n p l a c e i n p a r t s o f t h e beam and t h e s t r e s s - s t r a i n r e l a t i o n i n these p a r t s o f t h e beam a r e no l o n g e r g i v e n by t h e s t r e s s - s t r a i n curve assumed i n t h e theory.  44  TABLE 4 UNIT FUNCTIONS FOR A l 65S-T6 BEAMS (a) K = 0 m  n  -s  x-lO  3.8 3.9 4.0 4.2 4.4 4.6 4.8 5.0 5.3 5.6 6.0 7.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 2A.0 26.0 28.0 30.0 32.0 36.0 40.0 44.0 48.0 52.0 56.0 60.0 64.O 68.0 72.0 76.0 80.0 84.0 88.0 90.0 92.0 94.0 96.0 100.0 104.0  (b) K - 0.5 u.  n dm  12.10 12.42 12.73 13.31 13.84 14.32 14.75 15.13 15.64 16.08 16.57 17.47 18.06 18.78 19.19 19.46 19.64 19.77 19.87 19.95 20.02 20.08 20.15 20.20 20.26 20.37 20.46 20.55 20.63 20.71 20.79 20.86 20.94 21.00 21.06 21.11 21.16 21.20 21.24 21.26 21.28 21.30 21.31 21.33 21.35  22.99 24.21 25.43 27.83 30.11 32.26 34.27 36.15 38.77 41.16 44.02 49.84 54.30 60.79 65.30 68.78 71.56 73.72 75.54 77.20 78.81 80.41 82.16 83.85 85.61 89.22 92.79 96.36 100.22 104.27 108.49 112.84 117.30 121.52 125.44 129.29 133.19 136.88 140.49 142.19 143.82 145.31 146.64 148.80 150.02  185 1.50 200. 1 . 5 1 216. 1.53 247. 1.57 278. 1.62 308. 1.67 337. 1.73 366. 1.80 406. 1.87 444. 1.98 490. 2.17 590. 2.37 669. 2.80 3.20 788. 3.72 874. 941. 4.13 995. 4 . 6 2 1038. 5.11 1074. 5.34 1107. 5.30 1139. 5.10 1171. 4.69 1207. 4 . 4 1 1 2 4 1 . 4.25 1276. 3.93 1350. 3.77 1423. 3 . 6 1 1496. 3.37 1575. 3 . 1 1 1659. 2.94 1746. 2.81 1837. 2.71 1930. 2.68 2019. 2.75 2101. 2.80 2182. 2.76 2265. 2.75 2343. 2.79 2420. 2.82 2456. 2.89 2490. 2.97 2522. 3.20 2550. 3 . 6 1 2596. 4.47 2623. 14.89  10  3.8 3.9 4.0 4.2 4.4 4.6 4.8 5.0 5.3 5.6 6.0 7.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0 32.0 36.0 40.0 44.0 48.0 52.0 56.0 60.0 64.0 68.0 72.0 76.0 80.0 84.O 88.0 90.0 92.0 94.0  in?-  30.25 30.97 31.53 32.48 33.22 33.84 34.35 34.83 35.44 35.95 36.50 37.44 38.09 38.88 39.34 39.68 39.89 40.02 40.17 40.30 40.44 40.58 40.72 40.85 40.96 41.19 41.36 41.55 41.76 41.94 42.11 42.28 42.42 42.50 42.60 42.70 42.79 42.86 42.92 42.94 42.95 42.95  U  4.1,  in?  57.48 60.23 62.45 66.38 69.52 72.34 74.70 77.08 80.21 83.01 86.16 92.31 97.14 104.31 109.36 113.82 116.97 119.13 121.91 124.62 127.95 131.43 135.21 139.06 142.38 150.23 156.66 164.43 174.04 183.09 192.70 202.56 211.06 216.27 223.34 230.52 237.54 243.69 249.02 250.27 250.99 251.55  1159 1243. 1313. 1438. 1542.  I636.  1.20 I.25 1.31 1.37 1.46 1.54 1.60 1.66 1.74 1.86 2.09 2.29 2.64 2.98 3.30 3.57 4.35 4.36 3.76 3.34 2.93 2.71 2.54 2.56 2.41 2.38 2.32 2.02  1717. 1799. 1909. 2009. 2123. 2350. 2533; 2809. 3006. 3183. 3308. 3394. 3505. 3614. 3749. 3890. 4044. 4201. 4337. 4659. 4924. 5246. 5647. 1.89 6025. 1.86 6429. 1.80 6845. 1.82 7205. 2.06 7427. 2.17 7727. 1.98 8034. 1.96 3334. 2.01 8597o ,2.14 8826. 2.46 8879. 4.17 8910. 5.79 8934. 14.20  45 TABLE 4 U n i t FUNCTIONS FOR A l 65S-T6 BEAMS (c) K -J 3.3 3.9 4.0 4.2 4.4 4.6 4.3 5.0 5.3 5.6 6.0 7.0 8.0 10.0 12.0 14.0 16.0 13.0 20.0 22.0 24.0 26.0  28.0 30.0 32.0 36.0 40.0 44.0 48.0 52.0 56.0 60.0 64.0 68.0 72.0 76.0 80.0 84.0 88.0 90.0  -  a.  too UipS  43.40 49.52  50.33 51.66 52.59 53.37 53.95 54.53 55.24 55.33 56.42 57.42 53.11 53.93 59.49 59.91 60.14 60.27 60.47 60.65 60.87 61.08 61.30 61.50  61.66 62.02 62.26  62.55 62.38 63.16 63.44 63.70 63.91 64.00 64.15 64.29 64.42 64.52 64.60 64.61  91.96 96.26 99.46 104.93 103.93 112.43 115.14 118.01 121.65 124.36 123.30 134.73 139.93 147.32 153.43 153.37 162.39 164.55 163.27 172.03 177.09 132.44 183.25 194.23 199.15 211.25 220.52 232.49 247.36 261.91 276.92 292.29 304.31 311.02 321.24 331.74 341.38 350.50 357.54 353.35  (d) K  1.0  z  1.5  1*  duo  2967 1.13 3178. 1.17 3333. 1.22 3616. 1.28 3325. 1.36 4010. 1.45 4156. 1.51 4311. 1.55 I.64 4511. 1.77 4690. 2.02 4883. 2.21 5251. 5552. 2.51 6011. 2.80 6343. 2.99 6663. 3.17 6879. 4.12 7009. 3.34 7234. 3 . 0 2 .7462. 2.61 7769. 2.26 3095. 2.11 8450. 1.99 8821. 2.02 9121. 1.93 9869. 1.92 10445. 1.38 11192. 1.65 12156. 1.57 13041. 1.55 13991. 1.51 14969. 1.54 15768. 1.77 16164. 1.33 16819. 1.67 17494. 1.66 18146. 1.71 18702. 1.83 19157. 2.22 19209. 10.97  3.3 3.9 4.0 4.2 4.4 4.6 4.3 5.0 5.3 5.6 6.0 7.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 23.0 30.0 32.0 36.0 40.0 44.0 48.0 52.0 56.0 60.0 64.0 63.0 72.0 76.0 80.0 84.O 33.0  66.55 63.07 69.13 70.33 71.97 72.39 73.55 74.23 75.04 75.70 76.35 77.39 73.14 79.08 79.64 30.14 80.39 30.52 80.77 31.00 81.29 81.53 81.37 82.15 82.36 82.34 83.16 33.55 84.01 84.39 84.76 85.12 35.39 35.50 35.69 85.38 86.05 86.18 86.23  1 2 6 . 4 5 5610 1.09 1 3 2 . 2 8 6OO3. 1.12 136.47 6290. 1.17 143.43 6730. 1.22 143.34 7128. 1.30 152.51 7430. 1.39 155.53 7654. 1.44 153.94 7902. 1.43 163.10 8213. 1.56 166.72 3436. 1.70 170.44 3769. 1.96 177.24 9292. 2.14 182.32 9726. 2.40 1 9 1 . 3 4 D395. 2.65 197.50 10334. 2.75 203.91 11396. 2.89 207.31 H709. 3.91 209.97 11383. 3.46 214.64 12260. 2.60 2,22 219.45 12649. 226. 3 13199. 194 233.46 13738. 1.32 241.29 14427. 1.73 249.50 15101. 1.76 255.91 15628. 1.69 272.27 16979. 1.69 234.39 17935. 1.66 300.56 19333. 1.43 321.67 21102. 1.42 340.72 22706. 1.41 361.13 24432. 1.33 332.01 26206 1.40 393.57 27617. 1.61 4 0 5 . 7 6 28232. 1.70 419.13 29376. 1.51 432.97 30563. 1 . 5 0 446.23 31703. 1.54 457.30 32657. 1.66 4 6 6 . 0 7 33413. 2.05  46  TABLE 4 (e) Uips  3.8 3.9 4.0 4.2 4.4 4.6 4.8 5.0 5.3 5.6 6.0 7.0 8.0 10.0 12.0 14.0 16.0 13.0 20.0 22.0 24.0 26.0 23.0 30.0 32.0 36.0 40.0 44.0 43.0 52.0 56.0 60.0 64.0 63.0 72.0 76.0 30.0 34.0 88.0  34.70 86.62 37.93 90.01 91.34 92.42 93.15 93.93 94.34 95.53 96.27 97.37 93.16 99.13 99.79 100.36 100.64 100.77 101.07 101.35 101.72 102.08 102.45 102.30 103.06 103.67 104.06 104.55 105.13 105.61 106.09 106.54 106.33 107.00 107.24 107.47 107.68 107.34 107.96  UNIT FUNCTIONS FOR A l 65S-T6 BEAMS  K - 2.0  v 'o v  160.93 163.31 173.43 132.03 137.75 192.60 196.02 199.36 204.54 203.57 212.59 219.71 225.66 234.35 241.56 243.95 253.22 255.33 261.01 266.86 275.37 284.47 294.33 304.71 312.68 333.23 343.25 368.62 395.49 419.54 445.35 471.74 492.33 500.51 517.03 534.20 550.53 564.ll 574.60  (f)  u.  KIO-5  .9037 9719 10171 10931 11450 11395 12212 12572 13014 13397 13733 14472 15055 15961 16629 17368 17797 18015 18532 19175 20039 20967 21974 23040 23360 25990 27545 29669 32436 35020 37752 40553 42754 43630 45399 47242 49004 50462. 51594  K - 2.5  1.07 1.10 1.14 1.13 1.26 1.34 1.39 1.42 1.50 I.64 1.91 2.08 2.30 2.52 2.57 2.66 3.74 3.17 2.32 1.99 1.75 I.65 1.53 1.61 1.55 1.55 1.53 1.33 1.33 1.32 1.30 1.32 1.50 1.53 1.41 1.40 1.44 1.54 1.92  3.3 3o9 4.0 4.2 4.4 4.6 4.3 5.0 5.3 5.6 6.0 7.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0 32.0 36.0 40.0 44.0 43.0 52.0 56.0 60.0 64.0 68.0 72.0 76.0 30.0 84.0 83.0  fe  102.35 105.17 106.73 109.18 110.72 111.94 112.75 113.63 114.64 115.45 116.20 117.34 118.19 119.28 119.94 120.59 120.89 121.02 121.37 121.70 122.14 122.53 123.02 123.45 123.76 124.49 124.96 125.55 126.26 126.34 127.41 127.96 128.36 128.50 123.73 129.06 129.31 129.50 129". 64  195.42 13393 204.33 14326 210.50 14930 220.53 16063 227.16 16792 232.63 17407 236.45 17330 240.79 13321 245.93 13913 250.42 19424 254.73 19923 262.13 20793 268.51 21533 273.37 22709 235.63 23573 293.99 24533 293.64 25144 300.30 25406. 307.37 26202 314.23 27042 324.52 .28290 335.49 29633 347.37 31091 359.93 32639 369.44 33315 394.30 369OO 412.12 39123. 436.69 42201 469.31 46307 493.36 49933 529.57 53951 561.47 53024 586.08 61179 595.26 62357 614.92 64387 635.42 67530 654.92 70049 670.92 72119 683.12 73700  1.06 1.08 1.12 1.16 1.23 1.30 1.35 1.33 1.45 1.59 1.86 2.03 2.22 2.41 2.42 2.49 3.53 2.94 2.12 1.33 1.62 1.54 1.43 1.51 1.46 I.46 1.44 1.31 1.27 1.27 1.25 1.27 1.43 1.50 1.35 1.34 1.37 1.46 1.82  47  TABLE 4  UNIT FUNCTIONS FOR A l 65S-T6 BEAMS (h)  (g) K = 3 . 0 * 10  3.3 3.9  4.0 4.2  4.4 4.6  4.3 5.0 5.3 5.6  6.0  7.0 3.0 10.0 12.0 14.0 16.0 13.0 20.0 22.0  24.0 26.0  28.0 30.0  32.0 36.0  40.0 44.0 43.0 52.0 56.0  60.0 64.0 68.0 72.0 76.0 30.0 34.0 33.0  no  tops  '|K. 121.00 123.72 125.53 V  128.36  130.09 131.47 132.35 133.33 134.44 135.33 136.12 137.32 133.21  139.33  140.09 140.31 141.14 141.27 141.67 142.05 142.57 143.08 143.60 144.10 144.46 145.32  145.36 146.55 147.33 143.06 143.74 149.33 149.35 150.00 150.33 150.65 150.94 151.16 151.32  K = 1.330 u  u 229.90 240.36 247.51 259.13 266.57 272.77 276.89 281.72 237.42 292.27 296.37 304.64 311.35 321.39 329.70 339.03 344.05  346.21 353.74 361.70  373.66  386.51 400.41 415.14 426.21 455.31 475.99  504.76 543.12 577.17 613.78 651.19 679.34  690.00  712.82 736.65  759.27 777.72 791.65  13545 19325 20716 22191 23153 23963 24507 25148 25912 26566  27190 28253  29177 30639 31731 33042 33750 34055 35119 36243 37950  39736 41779 43393 45495 49712 52721 56923. 62566  67596 73029 73605 82891 34415 87341 91427 94333 97625 99731  xio  1.05  3.3  1.07 1.10  3.9  IK?-  60.33 61.76  4.0 4.2 4.4 4.6 4.3 5.0  62.73  5.6 6.0 7.0 3.0 10.0 12.0 14.0 16.0 13.0 20.0 22.0 24.0 26.0 1.41 23.0  68.94 69.57  1.14  1.20 1.27 1.32 1.34  1.41  1.54 1.82 1.93 2.14 2.32 2.29 2.35 3.44 2.75 1.93 1.71 1.53 1.46 1.43 1.39 1.40 1.33 1.26 1.23 1.23 1.21 1.23 1.37 1.44 1.30 1.29 1.32  5.3  30.0  32.0 36.0 40.0  44.0 43.0 52.0 56.0  60.0 64.0 68.0 72.0 76.0 80.0 1.40 34.0 1 . 7 3 88.0  64.31  65.33 66.25 66.88 67.53  63.31  70.60 71.33 72.25 72.79 73.26  73.51 73.64 73.37 74.08 74.35 74.61 74.37  75.13 75.32 75.76 76.05 76.41 76.82  77.17 77.51 77.34 73.09 73.19 73.37 73.54 78.69  73.32  73.91  114.72  120.03 123.39 130.37 134.94 133.89 141.33 145.02 149.01 152.49 156.12 162.30 168.26 176.54 182.51 133.59 192.36 194.52 193.87 203.33  209.53 216.11 223.25 230.72 236.61 251.52  262.67 277.42  296.57 313.92 332.50 351.51 366.69  373.55  335.35 393.55 410.75 420.99 429.17  4613  4942 5182 5594 5890 6150 6346  656O  6831 7070 7321 7790  8177 8772  9205 9649 9926 10084 10405 10735 11195 11685 12219 12779 13222 14343 15195 16319 17737 19122 20559 22035 23219  23755 24713 25715 26674 27430 28125  1.10 1.13 1.19  1.24  1.32 1.41 1.46 1.50 1.59 1.72 1.93 2.17 2.43 2.70 2.82 2.97 3.93 3.57 2.72 2.33 2.03 1.90  1.80  I.84 1.76 1.76 1.72 1.52 1.46 1.45  1.41 1.44 1.66 1.75 1.56 1.55 1.59 1.71 2.10  52  PART I I BEAM TEST RESULTS Two aluminum a l l o y beams were l o a d e d t o f a i l u r e t o see whether o r n o t t h e moment r e d i s t r i b u t i o n p r e d i c t e d by l i m i t d e s i g n t h e o r y t a k e s p l a c e .  A l s o t h e moments,  d e f l e c t i o n s and f a i l u r e c o n d i t i o n o f t h e beam t e s t s can be compared t o t h e p r e d i c t i o n s o f the i n e l a s t i c b e n d i n g t h e o r y . The arrangement, l o a d i n g and d e f o r m a t i o n measurement  loc-  a t i o n s a r e f u l l y d e s c r i b e d i n S e c t i o n 1,(1); (1) R e s u l t s o f Beam Test 1 Beam s t r a i n s and d e f l e c t i o n s a r e g i v e n i n Table 5 and F i g s . 13 and 14.  S y m m e t r i c a l l y l o c a t e d measurements  which were found t o be i n c l o s e agreement were  averaged.  Included w i t h the l o a d - s t r a i n curves i s the s t r a i n a t the e l a s t i c l i m i t determined  from T e n s i o n Test  1.  D u r i n g i n e l a s t i c d e f o r m a t i o n , t h e beam e x h i b i t e d c o n s i d e r a b l e creep w h i l e t h e d e f o r m a t i o n measurements were b e i n g made.  In order t o maintain a constant load while  measurements were b e i n g made, t h e d e f l e c t i o n o f t h e beam had t o be i n c r e a s e d a t t h e l o a d p o i n t s .  S t r a i n s and d e f -  l e c t i o n measurements were t a k e n a f t e r a p e r i o d o f t i m e when  53  most o f the creep had taken p l a c e . I n i t i a l l y Test Beam 1 d e f l e c t e d under l o a d .  elastically  A f t e r a l o a d o f about 17 k i p s ,  inelastic  s t r a i n s were observed i n the f l a n g e over the support and  (Fig.14)  the beam began t o d e f l e c t more r a p i d l y with an i n c r e a s e  of l o a d  ( F i g . 1 3 ) . As the l o a d i n c r e a s e d  the f l a n g e  beyond 17 k i p s  s t r a i n s over the support i n c r e a s e d  the remainder o f the beam remained e l a s t i c .  r a p i d l y while At a l o a d o f  about 30 k i p s l a r g e i n e l a s t i c s t r a i n began t o take place i n the f l a n g e s below the l o a d p o i n t s  ( F i g . 14) and the beam  began t o d e f l e c t very r a p i d l y with l o a d  ( F i g . 1 3 ) . At a  l o a d o f 32 k i p s , the f l a n g e o f the beam between the l o a d p o i n t and the outer  support buckled l a t e r a l l y .  The buckled  c o n d i t i o n s e t up a l a t e r a l f o r c e on one o f t h e outer p o r t s and the beam s l i d  o f f i t s support.  sup-  Unfortunately  t h i s was not the type o f f a i l u r e under study and t h e bendi n g f a i l u r e c o n d i t i o n was not reached d u r i n g the t e s t . Shear s t r a i n s a r e shown i n F i g . 14.  The gauges  f o r measuring shear s t r a i n were l o c a t e d i n the c e n t r e o f the web.  I f i t i s assumed that the centre o f t h e web i s  under a c o n d i t i o n o f pure shear, then the s t r a i n s were measured a t 45 degrees t o the beam a x i s are the p r i n c i p a l s t r a i n s and must, i n the case o f pure shear, be equal and  54  opposite.  The shear s t r a i n t h e n i s e q u a l t o t w i c e e i t h e r  of the s t r a i n s .  A l t h o u g h the s t r a i n s were f o u n d t o be  not q u i t e e q u a l , the shear s t r a i n has been determined from t h e i r average on the b a s i s o f pure s h e a r , t h a t i s by addi n g o p p o s i t e t e n s i o n and compression s t r a i n s .  Included i n  F i g . 14 i s the shear s t r a i n c o r r e s p o n d i n g t o a shear s t r e s s which i s \ ^  °f  t n e  e l a s t i c l i m i t i n tension.  The  shear  s t r a i n 4 i n c h e s from the c e n t r e s u p p o r t remains e l a s t i c throughout t h e t e s t whereas the shear s t r a i n 2 i n c h e s from the  s u p p o r t becomes c o n s i d e r a b l y l a r g e r when t h e s e c t i o n  comes under the i n f l u e n c e o f i n e l a s t i c bending.  TABLE 5  OBSERVED DEFORMATIONS - BEAM TEST 1  MICRO STRAINS 1.5 LOAD . KIPS 5 10 12.5 15 17.5 18.75 20 21.25 22.5 23.75 25 26.25  27.5 28.75  30 30.5 31.25 32.0  OVER SUPPORT 1320 2580 3240 3970 5050 6040 7860 10150  13200  UNDER LOAD 513  1070  1350  I63O 1920  2060 2270 2480  2760 3050 3390  3720  4180 4730  5480 665O 10050  DEFLECTIONS IN. SHEAR  IN. FROM SUPPORT  TENSION 1080 2150 2690 3220 3640 3850 4380 5160 6540  8060  10500  L IN.FROM SUPPORT  COMPRESSION 1020 2000  1052 2050 2540 3060 3660  2450 2860 3240 3440 3720 4130  4780 5320  4710  5820  5260 6170 7200 836O  4120  6200  6680  868 1724 2160 2580 2960  3160 3340 3540 3740 3920  4100  8040 8920  4260 4460 466O  11160 13840  4840 4980  7080  15660  5160  UNDER LOAD .082 .156  .193  .230 .269 .294 .328 .366  .415 .467 .525 .582  .642 .706  .768 .860 .974  1.108  L FROM CENTRE SUPPORT 0.45  .109 .205 .253 .302 .353 .383 .424 .472 .530 .594 .663 .730  .800 .875  .942 1.007  1.102 1.280  58  (2) R e s u l t s o f Beam Te3t 2 Up t o a l o a d o f 32 k i p s , Beam Test 2 behaved s i m i l a r l y to Beam Test 1. occurred and  I n e l a s t i c bending s t r a i n s  over the c e n t r a l support  first  at a load o f 17.5 k i p s  under t h e l o a d p o i n t s at a load o f 31 k i p s .  As the  l o a d i n c r e a s e d from 31 k i p s , the beam d e f l e c t e d more r a p i d l y w i t h l o a d and because o f the c o l l a r attachments f a i l u r e by f l a n g e b u c k l i n g was prevented. 3 4 . 1 k i p s the beam f a i l e d support.  At a l o a d o f  i n bending over the c e n t r a l  The f l a n g e on the t e n s i o n s i d e o f the beam  f a i l e d by necking and breaking a t e i t h e r s i d e o f the web. F i g . 15a shows the f l a n g e break. the c e n t r a l support  When the s t i f f e n e r s a t  were removed, i t was found t h a t neck-  i n g and breaking a l s o occurred, adjacent  t o one o f the  s t i f f e n e r h o l e s on the t e n s i o n s i d e o f the beam- ( F i g . 15b).  T h i s f a i l u r e adjacent  t o the h o l e could not be  seen d u r i n g t h e t e s t and i t was not known a t what  stage  of the l o a d i n g t h a t t h i s f a i l u r e took p l a c e . As i n Beam Test 1,  while m a i n t a i n i n g  the l o a d  to take measurements d u r i n g i n e l a s t i c deformations,  cons-  i d e r a b l e creep took p l a c e , and the measurements were taken a f t e r a p e r i o d o f time when creep had n e a r l y stopped.  59  Beam s t r a i n s and d e f l e c t i o n s are given i n Table 6.  L o a d - d e f l e c t i o n curves are given i n Fig.16;  s t r a i n s f o r computing the beam moments are shown i n F i g . 17;  and s t r a i n s under the l o a d s , over the support  and i n the web f o r shear are shown i n F i g . 18.  As i n  Beam Test 1, except f o r the gauges f o r determining moments, symmetrically located measurements were found to be i n very close agreement and were averaged f o r presentation. The l o a d - d e f l e c t i o n curves show two d e f i n i t e changes of slope; (1)  at a load of 17»5 k i p s when i n -  e l a s t i c bending occurs over the c e n t r a l support and, (2)  at a load of 31 k i p s when i n e l a s t i c bending begins  under the load.  In the l o a d - d e f l e c t i o n curves f o r both  beam t e s t s ( F i g . 13 and 16), up to a load of about 10 k i p s there i s a divergence from the usual e l a s t i c s t r a i g h t l i n e r e l a t i o n s h i p between load and d e f l e c t i o n . the l o a d - s t r a i n curves of F i g . 14,  Since  17 and 18 do not ex-  h i b i t t h i s divergence, i t must be a t t r i b u t e d to s e t t l e ment of supports.  In Beam Test 2, under both the load  point and at a distance 0.45L from the c e n t r a l support, the support settlement determined by extending the e l a s t i c d e f l e c t i o n curve to zero was 0.015  inches.  The t a b u l a t e d  r e s u l t s and the l o a d - d e f l e c t i o n curves g i v e the measured  i  60  deflections but elsewhere the support settlement w i l l be 'subtracted-before the d e f l e c t i o n i s given.  The deflection  under the load point was O.246 inches at a load of 17.5 kips 'compared to 0.758 inches at 31 kips and more than 1.52 inches when f a i l u r e occurred at 34.1 k i p s . The load-strain curves of F i g . 18 show that the flange s t r a i n under the load was well into the i n - • e l a s t i c region when f a i l u r e occurred over the central support.  The flange strains f o r determining themoments ;  i n the beam i n F i g . 17 will.be discussed i n d e t a i l i n the following section. F i g . 3a and Figs. 15a-c show the deformed cond i t i o n of the beam over the central support and under s  the load points a f t e r the s t i f f e n e r s have been removed. The deformed state of the beam i s discussed i n Section 111,(4) a f t e r the beam moments have been calculated and the i n e l a s t i c theory calculations made.  TABLE 6 OBSERVED DEFORMATIONS - BEAM TEST 2  MICROSTRAINS LOAD KIPS  5  10 12.5 15 17.5 20 22.5 25 27.5 28.75  30  31  32 33 33.75 34.15  SHEAR STRAIN UNDER 2 IN. LOAD 4 IN. FROM SUPPORT POINTS FROM SUPPORT SUPPORT OVER  1180  2230  2740 3230  3660  4400  6450  500  1010  1250 1500 1730  1980 2360 2830  3360 3660  4030  4490 5530  912 1300 2250 2670 3080  3500 4020  4540 5240  5750 6370  7160  7950  9070  954  1360 2310 2780 3270 -3930 .,  5590  3130 11730 L4170 16330 19640 22400  10240 F  (For L o c a t i o n s see F i g .6 . )  GAUGES BETWEEN LOAD POINTS AND OUTER SUPPORTS  208 417 516 625 752 903 1091 1369 1560 1690 1329  1945  2044 2140 2153 A  GAUGES BETWEEN LOAD POINTS AND CENTRAL SUPPORT  ©  ®  337  352 690  325 642  974  1024  1099 1210 1130  1279  0  ®  552 663  733  902 1036 1320  CD  228 450  220  446  213 420 526 615 716  560 670  764 337 1070  333 1009  1536  1326  1596  1266  1705  1731  1925 2025 2104 2173  1626 1750  2083 2184 2242  1335  I  I865 1976  L  DEFLECTIONS IN.  660 818  1150  1253 1044 799  932 750  1493  660 520  620 491  1846 1935  417  397  238 199  2025 2093  U  355  313 208  94  94  R  793 963  1108 1220  1194 935 739  605 472 370 269 179-  79  UNDER ..AT . 4 5 L FROM LOAD CENTRAL SUPPORT  .085 .155 .190 .225 .261  .302  .371 .471  .533 . 644 .712  .773 .379 1.125 1.53  .099 .197 .243  .239  .335 .338 .474 .596 .728  .793 .375 .943  1.043  1.208 1.53  E  IP  62 F I G . 15.  Test Scary? 2  id) P l a ^ e o\s<zv Support  5oppovf  (c) W ^ b Load  under Point  After  Failor-g.  66 (3) Moments During Beam Test 2 The moments o c c u r r i n g i n the t e s t beam at the approach of f a i l u r e should be found t o see i f r e d i s t r i bution of moments occurs according to the theory of l i m i t design. The measured s t r a i n s f o r determining the beam moments at each stage of the l o a d i n g are shown i n Fig.17 f o r 7 of the 8 gauges.  Gauge (£) l o c a t e d on the top of  the flange between the load point and the c e n t r a l was not working properly during the t e s t .  support  Although the  s t r a i n measurements at the outer gauges are reasonably c o n s i s t e n t with each other as are the s t r a i n measurements at the inner gauges, there i s as much as 10 per cent v a r i a t i o n between s i m i l a r l y l o c a t e d gauges.  I t would be ex-  pected that symmetrically l o c a t e d gauges would give s i m i l a r measurements, but such gauges as the outer gauges (J) and (£) l o c a t e d on the top flange do not give s i m i l a r measurements.  Therefore f o r determining the moments at the  inner and outer gauge l o c a t i o n s , the f o u r outer gauges and the three inner gauges are averaged.  F i g . 19 shows  the average s t r a i n at the. inner gauges and a t the outer gauges.  67  The and  moments a t t h e l o c a t i o n o f t h e i n n e r gauges  o u t e r gauges can be c a l c u l a t e d f r o m t h e u s u a l  elastic  theory, M  - eEZ  (11)  where £ i s t h e measured f l a n g e s t r a i n , E the modulus o f e l a s t i c i t y , and Z t h e s e c t i o n modulus. I f , f o r the present,  t h e l o a d s and c e n t r a l s u p p o r t a r e  assumed a c t i n g a t p o i n t s c r e a t i n g a moment diagram o f s t r a i g h t l i n e s , t h e moments throughout t h e beam can be determined from t h e moments a t t h e l o c a t i o n s o f t h e gauges. Modulus o f E l a s t i c i t y from Measured Beam D e f l e c t i o n s The  modulus o f e l a s t i c i t y needed f o r t h e mom-  ent c a l c u l a t i o n s was found i n S e c t i o n 1,(3) t o be q u i t e v a r i a b l e among t h e samples t e s t e d .  For t h i s reason i t  i s b e s t t o choose a modulus r e p r e s e n t i n g t h e m a t e r i a l i n the t e s t beam.  The measured d e f l e c t i o n s up t o f i r s t  yield  can be used t o determine E by means o f t h e e l a s t i c beam theory.  The modulus should  be determined as c o r r e c t l y as  p o s s i b l e and t h e r e f o r e , a term t a k i n g account o f deform3 a t i o n due t o shear f o r c e w i l l be i n c l u d e d . The  t e s t beam i s s y m m e t r i c a l about t h e c e n t r a l  support and o n l y one s i d e need be a n a l y s e d .  The moment  o v e r t h e c e n t r a l s u p p o r t w i l l be determined from compata-  68  b i l i t y r e q u i r e m e n t o f t h e d e f l e c t i o n due t o e l a s t i c bendi n g s t r a i n s and shear s t r a i n s .  Fi^_20  (a) L o a d i n g  L  I (b) B e n d i n g Conjugate Beam x 1 EI~ (c)  Bending  Deflection S B  S B  =  1 EI  = L  2  PL  L  2L  4 * 2 * 3  PL  L  11L - M. L  2L~  4  8  12  3  M ~  ]"7PL  G  EI [ l 2 ? Shear  Shear  2  (+ upwards)  3  Deflection The d e f l e c t i o n due t o shear s t r a i n c a n c l o s e l y  be determined by assuming t h a t a l l t h e s h e a r f o r c e i s d i s t r i b u t e d e v e n l y a c r o s s t h e web a r e a A .  I f V i s t h e shear  w  J  f o r c e t h e a n g l e change o f the web o f t h e beam i s V G i s t h e shear modulus o f e l a s t i c i t y .  Sv  =  V  L  A TT  4"  W  -  R w"  v  3L  4-  (12)  where  69  From s t a t i c a l c o n s i d e r a t i o n s , R  =  - M  P 4  V = 3P + M  and  s  IT"  L~  Expressing  s  IT  (12) i n terms of P and M  g  and  simplifying, £  =  v  M A~G  (+ downwards)  s  C o m p a t a b i l i t y at the outer support that  S=Sw ,  L2 T 7 PL ET ET [128  or  B  - Ms>  2  Solving f o r M ,  M  s  =  s  3  21PL 128  E  The v a l u e of  jo.  is  2(1 + M )  =  M A^G" s  (13)  T l + 3EI~1  L Now  requires  L  where M i s  2  A G1 W  Poisson's Ratio.  given i n the A l c o a S t r u c t u r a l Handbook  0.33EI L A 2  w  =•  2(1 + 0.33)  G  (90)  Expression  2  Due  = 0.00440  (13) reduces to M  to Bending  (14)  (1.421)  D e f l e c t i o n Under the Load P o i n t (a)  19.05  s  = 0.1619 PL  (15)  70  R e f e r r i n g t o F i g . 20, the d e f l e c t i o n under the load  point from.the tangent t o the c e n t r a l support i s = 1  3M  S  4  L  4  8  ' 44M L  = 1 EI  L  1  M  2  4  s  L  2L  4  12  -  1  3P1  2  16  - 3PL "  2  3  C  16 * 8 * 12  L  (15) and s i m p l i f y i n g ,  Substituting ^B  •  = 0.002687  PL  3  EI (b) Due to Shear  \  =  V  L  =T3P  A^G  4  LT  (c)  Total  Deflection  0.002687  +  M ~7  L  TJ  A^G4  s  =  + 0.228  EI Using ( 1 4 ) , S = 0 . 0 0 3 6 9 0  0.228 PL A^G  EI L A G 2  W  PL3 EI  (16)  D e f l e c t i o n at 0 . 4 5 L from the C e n t r a l Support (a)  Due to Bending R e f e r r i n g t o F i g . 2 0 , the d e f l e c t i o n a t  X  L  L'  4  12  71  0.45  SB  L is Mx  1  .45L x .451 - .45M x .45L x .45L - PL x ,45L x .45L  s  s  2  EI +  .45  2  3  PL x . 4 5 L x . 4 5 L + 1 PL x L x (.45  2  4  2 4  3  S i m p l i f y i n g and s u b s t i t u t i n g S B  =  .003887  PL EI  2  4  -  .08325)L| -*  4  (15) f o r M  s  3  (b) Due t o Shear $V  =  0.223  PL  -  A G W  Substituting  (c)  Total  R  R_  L  A G  5  W  = P  - M  4  ~  s  g i v e s S v = 0.210 PL A G VJ  Deflection  = PL f" .003387 + .210 EI 5  ~EI §  k  =  L2l^G  L  .004310  PL  3 ( 1 7  )  EI I f $ is L  the d e f l e c t i o n under the l o a d  i n inches,  72  equation  (16) g i v e s E f o r the t e s t beam a s , E  =  141.2  P  sr A l s o , i f %YL I 0.45  st  n  e  d e f l e c t i o n at a d i s t a n c e o f  L from the c e n t r a l support, equation  (17) g i v e s E  as E  =  184.1  P  For Beam Test 1, deflection  when P = 1 5 . 0 k i p s , the net  $ i s 0 . 2 2 0 inches and SK i s 0 . 2 8 9 i n c h e s L  g i v i n g E v a l u e s o f 963O k i p s / 2 and 9560 k i p s / i n  ectively.  For Beam Test 2, when P  net d e f l e c t i o n $  L  i n  2 resp-  = 1 5 . 0 k i p s , the  i s 0 . 2 1 0 inches and S  K  i s 0.274 inches,  whereupon E.has v a l u e s o f 1 0 , 1 0 0 k i p s / ^ 2 and 1 0 , 1 0 0 k i p s / \ respectively.  The value o f E = 1 0 , 1 0 0 k i p s / ^ 2 w i l l be n  used f o r determining moments i n Beam Test 2. I n c o n s i s t e n c y o f Measured Moments and Loads Using the procedure o u t l i n e d a t the b e g i n n i n g of the s e c t i o n , the moments over the c e n t r a l support and under the loads were c a l c u l a t e d .  These moments should  check s t a t i c a l l y with the l o a d a c t i n g on the beam.  The  73  c a l c u l a t e d moments and c o r r e s p o n d i n g l o a d s d i d n o t check s t a t i c a l l y and t h e f o l l o w i n g i n v e s t i g a t i o n i s made i n t o the s o u r c e s o f e r r o r and a procedure i s chosen f o r d e t erming t h e moments i n t h e beam. i  The f o l l o w i n g s o u r c e s o f e r r o r may cause a d i f f e r e n c e between t h e measured l o a d and t h e l o a d determi n e d from t h e c a l c u l a t e d beam moments. A.  C o r r e c t n e s s o f t h e Load The l o a d r e c o r d e d by t h e t e s t i n g machine was  c a l i b r a t e d t o be c o r r e c t by a p r o v i n g B.  ring.  C o r r e c t n e s s o f t h e Moment a t t h e L o c a t i o n o f  the S t r a i n Gauges C a l c u l a t e d by M (1)  = €EZ  The s e c t i o n modulus Z, a g e o m e t r i c  property,  was determined from measurements o f t h e c r o s s - s e c t i o n t o be 6.35 i n .  3  and t h e p o s s i b l e e r r o r cannot be v e r y (2)  great.  The measurements o f S e c t i o n I , (3) i n d i c a t e  t h a t t h e modulus o f e l a s t i c i t y E v a r i e s w i t h l o c a t i o n and t h e r e f o r e , i s n o t known a c c u r a t e l y a t t h e beam s e c t i o n where the gauges a r e l o c a t e d .  The modulus 10,100 k i p s / i . n  average v a l u e d e t e r m i n e d f r o m t h e beam d e f l e c t i o n s .  i s an Not  much d e v i a t i o n f r o m t h i s modulus would be e x p e c t e d a t any  74  beam s e c t i o n . (3)  The f l a n g e s t r a i n 6 was measured by e l e c t r i -  c a l r e s i s t a n c e s t r a i n gauges.  One source o f e r r o r i s  t h a t t h e gauges may not have been g l u e d c o m p l e t e l y t o t h e .beam s u r f a c e o r t h e y may not have been l o c a t e d  centrally.  The o t h e r i s t h a t t h e a c t u a l gauge f a c t o r i s d i f f e r e n t from t h e one l i s t e d by t h e m a n u f a c t u r e r .  The m a n u f a c t u r e r  l i s t e d an e r r o r o f - 1 . 5 p e r cent i n t h e gauge f a c t o r . Fig.  17 shows t h a t t h e r e was a v a r i a t i o n i n s t r a i n r e a d -  i n g s from s i m i l a r l y l o c a t e d gauges. C. C o r r e c t n e s s o f Moments Throughout t h e Beam Determined from Moments G i v e n a t t h e L o c a t i o n s o f S t r a i n Gauges. I t was assumed t h a t t h e l o a d s and r e a c t i o n s were c o n c e n t r a t e d a t p o i n t s and t h e r e f o r e t h e moment d i a gram f o r o n e - h a l f t h e beam c o n s i s t s o f two s t r a i g h t as i n F i g . 2.  lines  I n Beam Test 2 o n l y t h e c e n t r a l s u p p o r t  had a spread r e a c t i o n .  F o r any l o a d , t h e p o r t i o n o f t h e  moment diagram o v e r t h e c e n t r a l support i s rounded o f f but t h e r e s t o f t h e moment diagram remains t h e same.  The  e f f e c t o f r o u n d i n g o f f can be c o n s i d e r e d a f t e r t h e moments have been d e t e r m i n e d .  Sources o f e r r o r i n d e t e r m i n i n g t h e  moments i n t h e beam based on moments a t t h e gauge l o c a t i o n s are:  75  (1)  I n c o r r e c t l o c a t i o n of s t r a i n gauges,  l o a d i n g p o i n t s , supports, e t c . The l o c a t i o n s of l o a d i n g points and suppdrts were made c a r e f u l l y and the l o c a t i o n s of s t r a i n gauges were rechecked a f t e r the t e s t was over. (2)  Reaction from the c o l l a r  supports.  During the l o a d i n g of the beam, the c o l l a r supports needed t o prevent f l a n g e b u c k l i n g may have exerted some upward r e a c t i o n on the beam due to f r i c t i o n a l f o r c e between the flange of the beam and the v e r t i c a l member of the c o l l a r .  I f there were some v e r t i c a l r e a c t i o n from  the c o l l a r s the moment diagram would contain three instead of two s t r a i g h t l i n e s .  Although the beam had f a i l e d , i t  was s t i l l unbroken and could be loaded when placed on the. two outer supports only.  The beam was set up and loaded i n  t h i s way with the c o l l a r s attached.  A f t e r l o a d i n g the beam,  the c o l l a r s were loosened t o see i f there were any changes i n the beam d e f l e c t i o n measured by the d i a l gauges.  Since  no change was observed i t i s very u n l i k e l y that any r e a c t i o n would have come from the c o l l a r s during Beam Test 2. I f none of the above e r r o r s were introduced,  76  t h e n t h e moments c a l c u l a t e d would s a t i s f y t h e f o l l o w i n g conditions: (1)  The c a l c u l a t e d moments w i l l check  stati-  c a l l y w i t h t h e measured l o a d . (2)  During e l a s t i c deformations, the r a t i o of  the moment o v e r t h e c e n t r a l s u p p o r t t o t h e moment under the l o a d p o i n t w i l l agree c l o s e l y w i t h the r a t i o i n e d from t h e e l a s t i c beam  theory.  Some e r r o r , i f v e r y l i t t l e , each o f t h e s o u r c e s l i s t e d . e r r o r which i s l i k e l y  determ-  i s possible i n  However, t h e o n l y s o u r c e o f  t o be o f c o n s i d e r a b l e  the e r r o r i n t h e f l a n g e gauge r e a d i n g s .  magnitude i s  On t h e b a s i s  the e r r o r i s i n t h e f l a n g e gauge r e a d i n g s ,  that  beam'moments  w i l l be c a l c u l a t e d by t h e f o l l o w i n g procedure i n which con d i t i o n s (1) and (2) (1)  above a r e s a t i s f i e d .  The average s t r a i n from t h e o u t e r gauges  i s i n c o r r e c t by a f a c t o r k  Q  which v a r i e s t h r o u g h o u t t h e  test. (2)  The average s t r a i n from t h e i n n e r gauges  i s i n c o r r e c t by a f a c t o r k^ which a l s o v a r i e s t h r o u g h o u t the t e s t .  77  k  (3)  The r a t i o  i / / k  o i s assumed t o be c o n s t a n t  t h r o u g h o u t t h e t e s t and i s determined from t h e moments duri n g e l a s t i c d e f o r m a t i o n s which must be i n t h e same p r o p o r t i o n as t h o s e p r e d i c t e d by t h e e l a s t i c beam t h e o r y . (4*)< values of k  Once t h e r a t i o  k  Vk  0  i s determined, t h e  and k± a t each s t a g e o f t h e t e s t c a n be d e t -  Q  ermined by t h e e q u a t i o n o f s t a t i c s between t h e moments and the l o a d . Calculation  o f Moments The e l a s t i c s o l u t i o n i n s t e p (3) has a l r e a d y E q u a t i o n ( 1 5 ) g i v e s t h e moment a t t h e  been worked o u t .  c e n t r a l s u p p o r t as  0.1619  PL.  The moment under t h e l o a d  point, ML  = 3  PL -  16  3 Ms  = 0.0660  PL  4  T h e r e f o r e , t h e r a t i o o f t h e moment o v e r t h e support  M  S  t o t h e moment under t h e l o a d  ML  i s0.1619  or  2.45.  0.0660  M  33  5  V  3,4/  A  Mo Mi  Th e moment under the l o a d i s M^. The moment a t t h e o u t e r gauges M  Q  i s 34 67.5  M  L  or  0.504  M^.  i s 2.45  The moment at the c e n t r a l support M  M^.  From s i m i l a r t r i a n g l e s the moment at the i n n e r gauges M  i  is p.45  L  (11.375)  -  1~[ M  22.5  L  or 0 . 7 4 4 M  L  .  J  = 0.744  During e l a s t i c deformation, %  Mo"  = 1.477  (18)  0.504  From the s t r a i n measurements, M^ = k^f^EZ and M  = k £" EZ  Q  0  Using (18), k i = £.1.477 k Ci  (19)  0  where  € and €i.are s t r a i n measurements up t o a  the e l a s t i c l i m i t of beam deformation.  Values o f  l K  i/k  a  r  e  0  worked out i n Table 7 f o r corresponding l o a d s up t o 15«0 kips.  The r a t i o  k  i / k remains f a i r l y c l o s e to i t s average  O.962 d u r i n g e l a s t i c deformation and w i l l be assumed the same throughout  the t e s t . U s i n g v i r t u a l work, the equ-  p  a t i o n of s t a t i c s between the moments and the l o a d i s ,  3/4 L P * = 4M 4>- 3MS4> L  or  Now  ML  16 M  =  67.5  34  L  + 12M  k £ EZ 0  Q  0  a  =  3 PL  = I.986 k ^ E Z  (20) (21)  79  From s i m i l a r t r i a n g l e s i n F i g . 21, M  s  = -1.986 [ k £ E Z + k j ^ j E z f l 11.12$ • o  0  k £ EZ ±  ±  11.375 or  M  s  =  1.942 k e E Z  + 1.980 k-^-jEZ  M  s  =  1.942 k € E Z  + 1.980 k  o  0  o  0  ±  k £ EZ 0  ±  (22)  *o~ Since i / k  =  0.962, e q u a t i o n s (21) and (22) xvhen substituted  in k ; o  k  0  =  3 PL EZ  Once k  Q  i n t o (20) g i v e a s o l u t i o n  (23)  1 55.1£  0  + 22.34^  i s f o u n d , k^ i s found from ( 1 9 ) ; t h e moment  under t h e l o a d i s found from (21) and t h e moment over t h e c e n t r a l support i s found from ( 2 2 ) .  Table 7 g i v e s t h e s e  v a l u e s f o r each stage o f t h e beam t e s t . I n c a l c u l a t i n g t h e moments i t was assumed t h a t t h e l o a d s and r e a c t i o n s were c o n c e n t r a t e d .  I n t h e beam  t e s t , t h e c e n t r a l support was spread over 2 i n c h e s and, due t o r o u n d i n g o f f o f t h e moment diagram, t h e moment c a l culated M 'support.  s  i s g r e a t e r t h a n t h e teist beam moment over t h e A l s o t h e s t i f f e n e r s c a r r y an unknown p o r t i o n o f  the moment d i r e c t l y o v e r t h e c e n t r a l support and under t h e  80  l o a d s and a comparison o f M  and M-^ w i t h c a l c u l a t e d  s  b e n d i n g s t r e n g t h cannot e a s i l y be made.  The moments  w i l l t h e r e f o r e be c a l c u l a t e d a t t h e h o l e l o c a t i o n s t o e i t h e r s i d e o f the l o a d p o i n t s and c e n t r a l The  support.  h o l e c e n t r e l i n e s a r e l o c a t e d one i n c h t o e i t h e r  s i d e o f t h e l o a d p o i n t s and c e n t r a l support b u t s i n c e f a i l u r e a t t h e h o l e s took p l a c e near t h e o u t e r (Fig.  edges,  1 5 b ) , t h e moments w i l l be c a l c u l a t e d near t h e  o u t e r edges o f t h e h o l e s 1.2 i n c h e s t o e i t h e r s i d e o f the l o a d p o i n t s and c e n t r a l s u p p o r t .  The moments a t  the h o l e s a r e i n p r o p o r t i o n t o t h e moments M^ and M . g  (1)  Moment a t t h e h o l e s o v e r t h e c e n t r a l  M  (2)  sh  =  M  ~ "  1  ,  2  M  22.5  support;  s^  Moment a t t h e h o l e s under the l o a d  points  and, (a) towards the c e n t r a l support ( i n n e r MLhi  ~  ML -  1.2  holes)  (M + M ) L  s  (b) away from t h e c e n t r a l support ( o u t e r MLho  88  M  L -  1.2 ML 62.5  holes)  81  The  moments at the h o l e s M  , M  S H  shown i n F i g , 22.  given i n Table 7 and  and  L H I  ML are h o  F i g . 23 shows  the moment d i s t r i b u t i o n r a t i o Msn/M^i* Beam Moments at the Approach of F a i l u r e The i n F i g . 22  c a l c u l a t e d moments at the  of l i m i t  moment over the support M ^ should G  be  D  e  c  a  u  s  e  M L ^ Q at the outer h o l e s  of the beam where the a f f e c t the  strength  the l i m i t of e l a s t i c deformation, the s  was  c a l c u l a t e d moment under the l o a d whereas at a l o a d of 33.8 k i p - i n . and moment 283  288  not kips -  c a l c u l a t e d moment k i p - i n . and  (MLhi) was  The  the  86 k i p - i n . ; 294  calculated  load point i s shown to  reasonably c o r r e c t by the extent  the l o a d  did  k i p s , these moments were  k i p - i n . under the  curves ( F i g . 1 6 ) ,  234  kip-in. respectively.  l o a d seen i n F i g . 15c.  small and  At a l o a d o f 17.5  i n bending.  (M ^)  holes  i s at a s e c t i o n  shear f o r c e was  over the c e n t r a l support  design.  compared t o  the moment under the l o a d p o i n t at the i n n e r %ihi  holes  i n d i c a t e that r e d i s t r i b u t i o n of moments  took place as assumed by the t h e o r y The  stiffener  be  of y i e l d i n g under the  Moreover, the l o a d d e f l e c t i o n  v e r i f y that y i e l d i n g took place under  point. At a l o a d of 33.3  k i p s , the  c a l c u l a t e d moment  82  under t h e l o a d p o i n t s a t t h e o u t e r h o l e s (MLho^  w  a  s  317  k i p - i n . compared t o t h e c a l c u l a t e d f a i l u r e moment  296  k i p - i n . a t t h e h o l e s over the c e n t r a l  (M ^). s  support  The g r e a t e r moment c a p a c i t y a t t h e o u t e r h o l e s  i s due t o t h e absence o f h i g h shear f o r c e .  This i s  v e r i f i e d by t h e y i e l d i n g p a t t e r n i n F i g . 15c w h i c h shows t h a t o n l y t e n s i o n o r bending  deformation  occurred  around the h o l e on t h e t e n s i o n s i d e o f t h e beam. The f a i l u r e l o a d was 34.1 k i p s . moment o v e r the c e n t r a l support  The maximum  296 k i p - i n .  b e f o r e f a i l u r e a t a l o a d o f 33 k i p s .  occurred  Necking and  f r a c t u r e a d j a c e n t t o t h e h o l e i n F i g . 15b may have o c c u r r e d a t a l o a d o f 33 k i p s and caused t h e moment t o become s l i g h t l y reduced. was  C o n s i d e r a b l e shear  yielding  a s s o c i a t e d w i t h t h e f r a c t u r e a t t h e h o l e and a g a i n  v e r i f i e s t h a t shear f o r c e c o n t r i b u t e d t o f a i l u r e i n t h e beam.  TABLE 7  LOAD KIPS  AVERAGE FLANGE MICROSTRAIN OUTER GAUGES  CALCULATION OF MOMENTS DURING BEAM TEST 2  GAUGE CORRECTION FACTORS  INNER GAUGES  Ki  ®-®  Ko  Ki  CALCULATED MOMENTS KIP-IN.  UNDER LOAD  *o  5  10 . 12.5 15 17.5 20 22.5 25 27.5 28.75 30 31 32 33 33.75  219  433  538  644  753  883  1065 1321  1559  1693 1320 1923  2022  2114 2165  333 664  322  939 1119  0.957 0.963 0.966 0.962  1236 1210  997  763 628  494  395 290 195  90  INNER CENTRAL OUTER LOAD HOES LOAD SUPPORT HOLES M  €t  -  1.065 1.030 1.087 1.039 1.099 1.096 1.093 1.101 1.122 1.126 1.134 1.136 1.142 1.143 1.172  1.025 1.039 1.046 1.043 1.058 1.055 1.057 1.060 1.080 1.084 1.091 1.093 1.099 1.105 1.129  30 60 75 39' 105  123  149 135  223  243 263 273  294  309  323  AT STIFFENER HOLES  OVER  M s  73 146  132 219  253  236 308 315  Lho 29 59 73  33 104  121  327 328  146 132 219 239 253 273 289  329  317  322  324  325  330  304  MLhi 24 49 61 73 86  102 125 159 194 213  232  246 261  275 288  CENTRAL SUPPORT  DISTRIBUTION OF MOMENTS Msh M  Lhi  Msh 63 135 168  203  234  264 284  239 293 294 294 295 295  296  294  2.79 2.73 2.77 2.77 2.72 2.60 2.28 1.82 1.51 1.33 1.27 1.20 1.13 1.075 1.02  (2-  87  PART I I I THEORETICAL PREDICTIONS AND FURTHER OBSERVATIONS (1) Behaviour o f Test Beam 2 Up To F a i l u r e as P r e d i c t e d By The I n e l a s t i c Bending Theory One o f the purposes  o f the beam t e s t s  was t o check the p r e d i c t i o n s o f the i n e l a s t i c ing  bend-  theory with regard to the moments d e f l e c t i o n s and  the c u r v a t u r e a t t a i n e d at the s e c t i o n o f f a i l u r e .  The  t h e o r y and symbols have been d e s c r i b e d i n S e c t i o n 1,(2) and the u n i t f u n c t i o n s used  i n the theory are presented  i n S e c t i o n 1,(4). Up t o the l i m i t o f e l a s t i c  deformation,  the e l a s t i c beam theory i s the same as the i n e l a s t i c bending t h e o r y .  The e l a s t i c beam t h e o r y has a l r e a d y  been used t o p r e d i c t moments and d e f l e c t i o n s i n S e c t i o n 11,(3).  The l i m i t o f e l a s t i c deformation  when the extreme f i b r e reaches  3 6 . 3 kips/.; 2  Tension Test 1.  i s reached  s t r e s s over the c e n t r a l -  support  the e l a s t i c l i m i t observed i n  The corresponding e l a s t i c moment  the c e n t r a l support i s 231 k i p - i n .  over  The moment under the  l o a d p o i n t from the e l a s t i c theory  ( i n c l u d i n g a term  for  or 94 k i p - i n .  shear deformation)  i s 23l/  9  > K  From  88  equation kips.  (20) t h e l o a d a t t h e e l a s t i c l i m i t i s 15.5  I n t h e beam t e s t , t h e e l a s t i c l o a d l i m i t i s  increased  somewhat because;  (1) t h e r e a c t i o n a t t h e  c e n t r a l s u p p o r t i s s p r e a d , and (2) t h e s t i f f e n e r s o v e r the support c a r r i e s p a r t o f t h e moment i n t h e beam. The  d e f l e c t i o n s d u r i n g e l a s t i c d e f o r m a t i o n o f t h e beam  t e s t a r e n o t p r e d i c t e d by t h e e l a s t i c t h e o r y  since the  measured d e f l e c t i o n s were used t o f i n d t h e v a l u e o f E. Moments D u r i n g I n e l a s t i c D e f o r m a t i o n s The volved  e l a s t i c s o l u t i o n o f t h e t e s t beam i n -  s a t i s f y i n g a c o n d i t i o n of compatability,  that  i s making t h e d e f l e c t i o n a t t h e o u t e r s u p p o r t t o t h e t a n g e n t over t h e c e n t r a l s u p p o r t z e r o .  The i n e l a s t i c  s o l u t i o n i n v o l v e s t h e same p r o c e s s o n l y , i n s t e a d o f b e i n g e x p r e s s e d as a s i n g l e e q u a t i o n ,  the s o l u t i o n req-  u i r e s a t r i a l and e r r o r p r o c e d u r e . I f t h e r e were no s t i f f e n e r s on t h e t e s t beam, t h e i n e l a s t i c s o l u t i o n would i n v o l v e o n l y t h e u n i t f u n c t i o n s g i v e n i n Table k (h) f o r t h e s e c t i o n o f beam which does n o t change t h r o u g h o u t t h e beam length.  Unfortunately,  complicating factors.  t h e s t i f f e n e r s p r o v i d e two F i r s t l y , the s t i f f e n e r  holes  change t h e c r o s s s e c t i o n o f t h e beam and reduce t h e  89  bending c a p a c i t y l o c a l l y .  The h o l e s a l t e r t h e s t r e s s  c o n d i t i o n s i n the s u r r o u n d i n g beam and t h e i r cannot be e x a c t l y d e t e r m i n e d .  effect  Secondly, the s t i f f e n e r s  c a r r y an unknown p o r t i o n d f t h e b e n d i n g moment i n t h e beam i n excess o f t h e moment c a r r i e d by t h e I s e c t i o n . Because o f t h e s t i f f e n e r s and h o l e s assumptions must be made t o a r r i v e a t a s o l u t i o n from t h e i n e l a s t i c bending theory. Assumptions f o r I n e l a s t i c Bending (a)  Solution  Bending C o n d i t i o n over t h e C e n t r a l Support Not o n l y does t h e s t i f f e n e r c a r r y an un-  known p o r t i o n o f t h e bending moment over t h e c e n t r a l s u p p o r t , b u t a l s o t h e 2 - i n c h wide support p r o v i d e s an unknown d i s t r i b u t i o n o f r e a c t i o n .  For c a l c u l a t i o n i t  w i l l be assumed t h a t t h e beam c u r v a t u r e i s c o n s t a n t f o r a d i s t a n c e o f 1.2 i n c h e s t o e i t h e r s i d e o f t h e beam centre l i n e .  T h i s c o r r e s p o n d s t o t h e o u t e r edges o f  the s t i f f e n e r h o l e s where f a i l u r e was t a k i n g p l a c e d u r i n g t h e beam t e s t . (b)  Bending C o n d i t i o n a t t h e H o l e s Because o f t h e reduced s e c t i o n , t h e o u t e r  90  f i b r e s t r a i n o r curvature  a t the h o l e s f o r . a n y  o f t h e moment i s i n c r e a s e d from the curvature s e c t i o n c o n t a i n i n g no h o l e s .  value o f beam  I f the h o l e s were l o c -  ated a t beam s e c t i o n s where o n l y e l a s t i c bending occu r r e d , then the e f f e c t o f the h o l e s i n determing d i s t r i b u t i o n o f moments would be o f secondary magnitude. However, i n the beam t e s t , c o n s i d e r a b l e i n g occurred  i n e l a s t i c bend-  at the h o l e s and the e f f e c t o f the h o l e s  i s no l o n g e r secondary.  From Table 4(h) the c a p a c i t y  of the beam i s reached when the f l a n g e s t r a i n i s 9 . p e r cent and the value o f the u n i t f u n c t i o n m i s 78.92 kips/j_ 2 n  The moment c a p a c i t y o f the beam s e c t i o n i s  319 k i p - i n . from equation  (3).  ent c a p a c i t y i s approximately  At the h o l e s , the momreduced by the moment o f  the t e n s i o n s t r e s s l o s t at the hole (5/8 i n . diameter) on the t e n s i o n s i d e o f the beam or 2 [(1/4 x 5/8) 42] or 13 k i p - i n .  For any value o f the f l a n g e  strain  given i n the t a b l e o f u n i t f u n c t i o n s , the moment a t the h o l e s w i l l be assumed  319 - 13 m o r 0.96m, where 319  m i s the u n i t f u n c t i o n f o r determining  deformation  con-  ditions. (c)  Bending C o n d i t i o n Under the Load P o i n t . The  s t i f f e n e r s at the l o a d p o i n t s c a r r y a  p o r t i o n o f the beam moment.  In p r e p a r i n g the s t i f f e n e r  91  for  t h e second beam t e s t , t h e h o l e s were o v a l l e d so as  t o prevent t h e s t i f f e n e r s from c a r r y i n g h o r i z o n t a l f o r c e s and consequent h i g h moments (See S e c t i o n 1,(1)). The  deformation  o f t h e two upper h o l e s shown i n F i g . 1 5  however, c l e a r l y i n d i c a t e s t h a t a h o r i z o n t a l f o r c e was c a r r i e d by t h e b o l t s .  A l t h o u g h t h e h o l e s were o v a l l e d ,  the b o l t c r e a t e d a dent on t h e s t i f f e n e r edge and t h i s p r o v i d e d a support f o r h o r i z o n t a l f o r c e .  The magni-  tude and d i r e c t i o n o f t h e f o r c e s a c t i n g on t h e b o l t s are d i f f i c u l t t o e v a l u a t e and hence t h e moment c a r r i e d by t h e s t i f f e n e r cannot be determined c l o s e l y .  For c a l -  c u l a t i o n i t w i l l be assumed t h a t t h e b o l t s b e a r t o t h e i r c a p a c i t y i n the v e r t i c a l d i r e c t i o n o n l y .  The c a p a c i t y  o f each v e r t i c a l row o f b o l t s i s [2(6) + d] k i p s  o r 20  k i p s from the b o l t c a p a c i t i e s worked out i n S e c t i o n The  moment c a r r i e d by t h e s t i f f e n e r under t h e l o a d  i s therefore  1,(1). points  assumed t o be 20 x 1 k i p - i n . o r 20 k i p - i n . The  e f f e c t o f t h e h o l e s extends past t h e  h o l e b o u n d a r i e s and t h e b e n d i n g c a p a c i t y o f t h e beam s e c t i o n without  s t i f f e n e r d i r e c t l y under t h e l o a d  p o i n t s w i l l be reduced a l t h o u g h holes.  n o t as much as a t t h e  F o r c a l c u l a t i o n , i t w i l l be assumed t h a t t h e  r e d u c t i o n o f moment c a p a c i t y i s l / 2 t h e r e d u c t i o n a t the h o l e s .  The moment i n t h e beam s e c t i o n w i t h o u t  stiff-  ener w i l l be assumed as 0.98 m where m i s t h e u n i t f u n c -  92 t i o n f o r determining  deformation  conditions.  A s o l u t i o n w i l l be made f o r d i f f e r e n t bend ing curvatures  (6 ) over t h e c e n t r a l support up t o and  i n c l u d i n g the f a i l u r e curvature. by assuming a f l a n g e s t r a i n 62  a t  The s o l u t i o n i s found the inner holes near  the l o a d p o i n t and s e e i n g i f t h e d e f l e c t i o n a t t h e end  i s zero.  Fig.  21+ Assumed Moments f o r I n e l a s t i c Bending Calculation  P a r t (1) The  a n g l e change a t t h e h o l e s o v e r t h e c e n t r a l  support ( p o i n t (T)), 4  = 2(1.2) g.„  h The  =  0.4221.€o  5.686  d e f l e c t i o n o f p o i n t ® w i l l be n e g l i g a b l e bec-  ause o f the spread  support.  93  P a r t (2) A value of the flange s t r a i n £ is assumed a t t h e i n n e r h o l e s  0 96rn,  2  (point 0  .  Because o f t h e h o l e s ,  the moment a t point©is g i v e n by O.96 mg where mg i s t h e u n i t f u n c t i o n c o r r e s p o n d i n g t o £2 X  "  20. lmm +m 0  2  For any beam o f l e n g t h X* h a v i n g c o n s t a n t  shear  f o r c e V ,, AM  V Substituting  F o r p a r t (2)  AVL =  Aw = V  1 A^Am,  A  2  X  = 2x  w  (24)  0  E~m  2(20.1) 5.686(0.96)(m  Q  + m) 2  = 7.365 m + m2 0  Angle change over p a r t (2) i s from e q u a t i o n ( 4 ) ,  4;  7.365 m  (n^i 0  +  n^)  m2  where t h e s u b s c r i p t t- r e f e r s t o the v a l u e o f n c o r r e s p o n d i n g t o O.96 m.  The d e f l e c t i o n o f p o i n t (2)from  94 the t a n g e n t a t © i s from e q u a t i o n  (21c)  of reference  (2). = h (^Y 2 MTV  §2  (U  Substituting for A  S2  w  Q  U  and x  = i$4.2 (u*i (tn  + 0  o l  }  ^xK"  )  n  7  and u s i n g  +  + n? )  +  ol>  £\ and  f o r o and o l ,  + 148.0 roz ( : n a - y 2 ) n  (m  2  2  0  + *> ) 2  2  P a r t (3) moment i n t h e beam a t p o i n t (3) i n c l u d i n g t h e  The  s t i f f e n e r i s g i v e n by, m,_  The  __  0 . 9 6 m 2 + 1 . 2 ( 0 . 9 6 ) (m  =  + m)  Q  2  = 1.017  m+ 2  0.0573 m  moment c a r r i e d by t h e s t i f f e n e r i s 20 k i p - i n .  I n terms o f t h e u n i t f u n c t i o n m, t h e moment i s g i v e n by 20  or 4.95 k i p / i n ?  The moment a t p o i n t Q) i n  1.421(2.343) the beam w i t h o u t 0.98 m  3  m^  =  s t i f f e n e r i s g i v e n by 0 . 9 3 1.017  1.038 m  = I Z  2  m  2  +  +  0.0573 m  0.0585 m  From (24) A V  w  Q  - 4.95  - 5-05  0  =  where,  (25)  2(1.2) 5.686(1113-1112)  0.4222 m^-n^  2  95 ^  =  1.2  m  3  Angle change o v e r p a r t 4>3  (3),  0.4222 (n3-n2)  =  m^ -nig d e f l e c t i o n o f p o i n t (J) from t h e t a n g e n t a t (2)  The  i s from e q u a t i o n ^3  =  (21b) o f r e f e r e n c e ( 2 ) ,  0.507 (U3-U2)  - 0.507m (n -n2)  <"V»3>  < 3" 2'  2  3  m  3  m  2  P a r t (4) The moment a t p o i n t ( 4 ) i s g i v e n by 0.96 m^ where, 0.96 m = 4  66.3 m 0775  L  = 66.3  (1.017 m + 0.0573 2  m) 0  6TT5  014 « 1.040 m2 + 0.0586 m  (26)  0  2(1.2) 5.686(1114-1113) 1.2m/  f  1114-1113 0.4222 1114- 1113  (n^-n^)  = 0.4222 1*14-1113  96  The d e f l e c t i o n o f p o i n t (4) from t h e tangent a t Q) i s %  K  =  0.507(U4-U3) (1114-1x13)  Part  -  0.507m4(n4-n3) (m^-m^)2  2  (5) From (24), A  w  ~\T  = 66.3  = 24.3  2.843(0.96m/^ m4  The d e f l e c t i o n o f p o i n t (5) from the  tangent a t p o i n t (£) i s from  e q u a t i o n (7)«>  - 1679  '5  V  m^  where the s u b s c r i p t ^ r e f e r s t o the v a l u e o f Lic o r r e s p o n d i n g to O.96 m^. The t o t a l d e f l e c t i o n a t the end o f the support t o the  tangent over the c e n t r a l support due to bending i s ,  S =S  2  +  *3 V% +  +  <M^.8) +-<*> (68.7) +4 (67.5) + 4 ( 6 6 . 3 ) 2  3  4  S u b s t i t u t i n g the a p p r o p r i a t e v a l u e s i n terms o f u n i t f u n c t i o n s and r e a r r a n g i n g ,  97  M  =  -  167911*4 V  +  0«507(U4-U3) + 0.507(u -U2) + 1 5 4 . 2 ( t o + Uig) (m -m ) (m m ) (m m ) 3  2  4  (g) -  3  r  (b)  a  2  Q+  (c)  4  3  (m-j-m )*"  2  2  (h)  ^ ( ^ i  2  ~~  0  +  (a)  Q  + m  X  3  m-  2  3  m  2  2  m^-  2  0  m  3  (27)  (j)  A l l other f u n c t i o n s can be determined  value of £  ^  - 23.5 ( n - n ) - 28.0 (114-113)  (f)  ^2 by equations  1  2  0  (k)  There are only two unknowns i n equation and 6 .  1  (e)  506 ( n ^ x - n ^ ) m  -  (m +m ) -  (i)  37.56  2  (d)  0.$07m^(n -n^) - 0.507m^(n -n ) + 148 (m^-m^) -  +  2  (27), €.  a  from € and Q  (25) and (26) and Table 4(h). . For any  o r curvature over the c e n t r a l support, t h e r e  i s only one v a l u e of £2 which w i l l make S o f e q u a t i o n (27) equal to zero.  The value o f £-2  e r r o r process.  S o l u t i o n s o f equations  i  s  found by a t r i a l and (27) are c a l c u l a t e d  i n Table 3 f o r t h e f a i l u r e curvature and 3 o t h e r c u r v a t u r e s at the c e n t r a l support.  The moments at the s t i f f e n e r h o l e s  c a l c u l a t e d from the theory are given i n Table 10.  The mom-  ents are p l o t t e d with the moments c a l c u l a t e d from the beam t e s t i n F i g . 22 and t h e corresponding moment r e d i s t r i b u t ions are p l o t t e d i n F i g . 23.  Values o f t h e t h e o r e t i c a l  l o a d s and moments when f a i l u r e occurs are presented i n  98  Table 11 with the t e s t  values.  Up t o a load of about 27 k i p s the t h e o r e t i c a l and t e s t moments agree very c l o s e l y .  From then  until  f a i l u r e t h e r e i s an i n c r e a s i n g d i v e r g e n c e . Failure, i s p r e d i c t e d by the theory a t a l o a d o f 3 3 . 2 k i p s when the moment r e d i s t r i b u t i o n given by the r a t i o M h s  i s 1.13  MLhi compared t o the t e s t f a i l u r e l o a d 34.1 k i p s and a r a t i o o f l e s s than 1 . 0 2 .  Mgh  The f a i l u r e moment at the h o l e s  MLhi adjacent to the c e n t r a l support the t h e o r y i s g r e a t e r than moment.  (Msh) 3-06 k i p - i n .  296 k i p - i n . , the t e s t  As p o i n t e d out i n S e c t i o n I I , ( 3 ) , h i g h  f o r c e reduces  from failure  shear  the bending moment c a p a c i t y of the beam and  i s i n s t r u m e n t a l i n the divergence between the t h e o r y and the t e s t moments beyond 27 k i p s . D e f l e c t i o n s During I n e l a s t i c  (27) the d e f l e c t i o n under the l o a d  From equation is  5  L  Deformations  = 0.507(U -U2) (m-^-mg)  + 154.2(u +UV2) (m + r r ^ )  (c)  (d)  3  2  + 148m (n -n^2) 2  U  yl  + 0  3  2  +  9<>0 £  Q  + 8.8 ( n ^ - n ^ ) 0  (m)  2  (h)  m+  2  3  2  0  m )2  (e)  0.507m (n -n ) (m-j-rr^)  yl  m  2  (n)  (29)  99  The d e f l e c t i o n a t 0 . 4 5 L from the c e n t r a l s u p p o r t ( p o i n t ® ) i s ^ e q u a l t o t h e d e f l e c t i o n a t (|) t o ' t h e t a n g e n t a t © minus 4 9 . 5 (<h.-4 y +<  Now^ =  - 24.3 m  and S  (n^-n ) k  4  = 2.343 T2.43\ u  = l679u  2  kk  k  • mi >k  =  +  l 6 7 9 u  m4  k 2  mo  (o)  (p)  20.9(n -n ) 3  m  ^4'  20.9^o ~ 364.6 ( n y j - n  1203(11^-0  +  2  (q) 20.9(n4-n )  3"" 2 m  (s)  k  3  +  m  2  (r)  (30)  m4-m  3  (t)  These d e f l e c t i o n s a r e c a l c u l a t e d i n T a b l e 9 f o r the f a i l u r e  c u r v a t u r e and t h e t h r e e o t h e r c u r v a t u r e v a l u e s  over t h e c e n t r a l support used i n t h e p r e v i o u s i n e l a s t i c b e n d i n g s o l u t i o n . D e f l e c t i o n s a r e l i s t e d i n T a b l e 10 and p l o t t e d w i t h the net t e s t d e f l e c t i o n s i n F i g . 25.  Values  o f t h e t h e o r e t i c a l and t e s t d e f l e c t i o n s a t f a i l u r e a r e g i v e n i n Table  11.  The l o a d - d e f l e c t i o n curves from t h e i n e l a s t i c b e n d i n g t h e o r y shown i n F i g . 25  f o l l o w t h e same shape as  100  the d e f l e c t i o n c u r v e s from t h e t e s t measurements. deflections  The  from t h e t h e o r y however, a r e 10 t o 20 p e r  cent l e s s than t h e t e s t d e f l e c t i o n s .  One r e a s o n f o r  t h i s i s that the t h e o r e t i c a l d e f l e c t i o n s the d e f o r m a t i o n due t o shear f o r c e .  do n o t i n c l u d e  At f a i l u r e t h e de-  f l e c t i o n under t h e l o a d was more t h a n 1.52 i n c h e s  com-  pared t o t h e computed t h e o r e t i c a l d e f l e c t i o n o f o n l y 1.01 i n c h e s . curvature  T h i s means t h a t a t f a i l u r e , t h e i n e l a s t i c  over t h e c e n t r a l support c o n t r i b u t e s most t o  the d e f l e c t i o n under t h e l o a d and i n d i c a t e s curvature  that the  a t f a i l u r e was g r e a t e r t h a n p r e d i c t e d by t h e  i n e l a s t i c bending  theory.  101  2) Moments and Deflections Predicted By The Theory Of Limit Design The p l a s t i c collapse load i s given by p l a s t i c hinges occurring at the holes adjacent to the central support (M ^) and at the holes under the load point s  nearest to the outer support  (ML^O)*  r  ^  ie  p l a s t i c moment  of the f u l l beam section i s 295 k i p - i n . and subtracting 13 k i p - i n . f o r the holes, the p l a s t i c moment at the hinges i s 282 k i p - i n . P  21.3 P © = M (l+1 + 22.5)6 66.3 p  — — •  ,  3 1 . 0 kips The moment under the load i s 67.5 M  p  and the  6~673  moment at the inner holes ( M T ^ ) i s'67.5 M L66.3  or 0.898 Mp.  p  -  1.2 (67.5 + 2 0 7 T ^ 66.3  Therefore, the l i m i t design theory predicts  that the r a t i o M h s  is  1  or 1 . 1 1 when f a i l u r e takes  place. The deflection just prior to collapse occurs when there i s a p l a s t i c hinge at the holes over the central support and a p l a s t i c moment just formed at the  AM  102  outer h o l e s under the load p o i n t M  P !l_]!  D e f l e c t i o n Diagram  1 4 >  -  Z  l  4>88.8  5  " , l =1 EI  4>  =  Conjugate  1  1  —-—  Beam x _l m  663"  38.8  Mp(83.8)  66.3  16.75 M  m  EI - 155.1  M  p  21.3  (67.5+2 x  21.3)  66.3  3  p  EI From e l a s t i c bending under the load i s 431Md e f l e c t i o n at p o i n t ©  or  theory, the d e f l e c t i o n  0.631 i n c h e s . S i m i l a r l y the  i s 599M  p  or  O.878 i n c h e s .  ,EI  The l o a d at which the f i r s t  p l a s t i c hinge  occurs  at the h o l e s over the c e n t r a l support i s found by the e l a s t i c theory i n S e c t i o n I I , (3) to be 2 0 . 9 k i p s . The moments and d e f l e c t i o n s p r e d i c t e d by the t h e o r y o f l i m i t d e s i g n are p l o t t e d i n F i g s . 22 and 23 with the t e s t v a l u e s and the v a l u e s from the i n e l a s t i c bending t h e o r y .  Values o f the moments and d e f l e c t i o n s  when the mechanism c o n d i t i o n i s reached are g i v e n i n  103  Table 11 w i t h t h e f a i l u r e v a l u e s from t h e t e s t and t h e i n e l a s t i c bending The  theory.  s i g n i f i c a n t d i f f e r e n c e s between t h e t e s t  moments and t h e l i m i t d e s i g n moments a r e : (1)  The t h e o r y o f l i m i t d e s i g n assumes no  s t r a i n h a r d e n i n g which accounts f o r t h e b u i l d up o f moment a t t h e c e n t r a l support beyond t h e p l a s t i c moment. (2)  The t h e o r y o f l i m i t d e s i g n p r e d i c t s a  p l a s t i c h i n g e under t h e l o a d a t t h e o u t e r h o l e s whereas i n e l a s t i c d e f o r m a t i o n was more predominant a t t h e i n n e r holes.  The presence  h o l e s reduced-the moment.  o f h i g h shear f o r c e a t t h e • i n n e r  c a p a c i t y o f t h e beam t o c a r r y bending  The l i m i t d e s i g n t h e o r y p r e d i c t s t h a t t h e mom-  ent a t t h e i n n e r h o l e s approaches 90 p e r cent o f t h e moment a t t h e c e n t r a l support whereas t h e t e s t i n d i c a t e s t h a t t h i s r a t i o was almost 100 per cent a t f a i l u r e .  10 ;4  (3) C o n f i g u r a t i o n o f t h e Test Beam A f t e r F a i l u r e I t i s i m p o r t a n t t o make complete o b s e r v a t i o n s o f t h e t y p e o f f a i l u r e and t h e i n e l a s t i c o r permanent d e f o r m a t i o n s t h a t took p l a c e i n t h e beam.  Failure  took p l a c e o v e r t h e c e n t r a l support i n t h e form o f f r a c t u r e i n t h e t e n s i o n f l a n g e and f r a c t u r e a d j a c e n t to one o f t h e s t i f f e n e r  holes.  The c o n f i g u r a t i o n a f t e r  f a i l u r e i s shown i n F i g . 3a and 15. Y i e l d i n g and d e f o r m a t i o n o f t h e h o l e s a r e q u i t e pronounced over t h e c e n t r a l s u p p o r t .  The s t i f f -  ener p r e v e n t e d much y i e l d i n g i n t h e s t i f f e n e d beam p o r t i o n - t h a t p o r t i o n o f t h e web e n c l o s e d by t h e s i x h o l e s i n F i g . 15b. 15b,  A s l i d i n g y i e l d can be seen i n F i g .  s t a r t i n g w i t h t e n s i o n y i e l d i n g extended i n t h e  f l a n g e above t h e s t i f f e n e r  and changing i n t o shear  i n g down through t h e o u t e r edges o f t h e h o l e s .  yield-  The  shear s l i d e c o n t i n u e s combining w i t h compression i n t h e l o w e r web, and t h e l o w e r f l a n g e i s bent l o c a l l y .  The  b e n d i n g o f t h e l o w e r f l a n g e was a g g r a v a t e d by l o c a l b u c k l i n g a c t i o n due t o t h e compressive f o r c e s i n t h e flange.  The presence o f h i g h shear f o r c e i s demonstrated  by t h e n o t i c e a b l e shear y i e l d through t h e row o f s m a l l  10 5  h o l e s on e i t h e r s i d e o f t h e c e n t r a l s u p p o r t .  Also the  e x t e n t o f y i e l d i n g i n t h e f l a n g e i s shown i n F i g . 15a where a c o n s i d e r a b l e l e n g t h o f t h e f l a n g e has c o n t r a c t e d l a t e r a l l y due t o P o i s s o n e f f e c t o f t h e t e n s i o n deformations. Although hole deformation occurs i n the upper h a l f o f the web which was i n t e n s i o n , t h e r e i s no h o l e d e f o r m a t i o n i n t h e l o w e r h a l f o f t h e web which was i n compression.  I f compression  d e f o r m a t i o n were  l a r g e w i t h i n t h e s t i f f e n e d beam p o r t i o n , then t h e two lower h o l e s would have deformed towards each o t h e r . Therefore, l i t t l e  compression  y i e l d i n g took p l a c e over  the c e n t r a l support w i t h i n t h e s t i f f e n e r .  Furthermore,  h i g h b e a r i n g o f t h e beam web a g a i n s t t h e s t i f f e n e r due t o P o i s s o n e f f e c t o f compression  d e f o r m a t i o n i n t h e web,  can o n l y be seen i n t h e web o u t s i d e t h e s t i f f e n e d beam. The o v a l l i n g o f t h e two upper h o l e s on t h e t e n s i o n s i d e o f t h e beam can be a t t r i b u t e d t o a combina t i o n o f t e n s i o n d e f o r m a t i o n and shear d e f o r m a t i o n o r s l i d i n g a c t i o n i n t h e web.  The d e f o r m a t i o n o f t h e m i d d l e  h o l e s i s a t t r i b u t e d m o s t l y t o t h e shear s l i d i n g  action.  F i g . 15b shows t h a t t h e t o p o f t h e s t i f f e n e r was b e a r i n g h e a v i l y on t h e bottom o f t h e upper f l a n g e .  This  indi-  106  c a t e s t h a t t h e s t i f f e n e r f a i l e d t o h o l d due t o e x cessive bearing o f the b o l t s against the s t i f f e n e r . Y i e l d i n g under t h e l o a d p o i n t i s shown i n F i g . 15c by t h e p e n c i l s h a d i n g .  Yielding at the inner  h o l e s under t h e l o a d p o i n t i s s i m i l a r b u t n o t as p r o n ounced as t h e y i e l d i n g over t h e c e n t r a l s u p p o r t .  At  the o u t e r h o l e s , t e n s i o n y i e l d i n g o c c u r s above and below t h e l o w e r h o l e on t h e t e n s i o n s i d e o f t h e beam. The  shear f o r c e a t t h e o u t e r h o l e s was much s m a l l e r  t h a n a t t h e i n n e r h o l e s and o v e r t h e s u p p o r t . The  deformed beam i n d i c a t e s t h a t t h e presence  o f h i g h shear f o r c e r e d u c e s t h e b e n d i n g c a p a c i t y and a l s o t h a t t h e moment a t t h e i n n e r h o l e s under t h e l o a d was almost e q u a l t o t h e moment a t t h e h o l e s o v e r t h e support when f a i l u r e  occurred.  TABLE  8.  C A L C U L A T I O N OF I N E L A S T I C BENDING S O L U T I O N F O R T E S T " B E A M 2.  Units:  THEORY  £ - i n / i n . x 10"^; m, n and u - k i p s / j _ x 10'^; d e f l e c t i o n - i n . x 10 2  n  €  0  mo  2  e  m2 2 ^2 (2) n  (1) 90 78.92  251.5 4350  48 76.82 196.7 10240  .20 73.87 165.4 7976  8 71.33 149.95 6896  5«6 68.94 152.5 7070 6.0 69.57 156.1 7321  n^2 (3)  my **3 3  U  (4)  71.13 138.6 167.5 6129 812 4 71.78 141.4 17 2.3 6318 8470  1*4  114 ^4  (a)  76.3 2 273.7 3377 160 3 8 9660 76.97 303.6 3377 18 3 20 10436  3o7 60.42 58.7 5 108.6 100.0 115.0 4260 3770 4632  65. 60 13 5.9 1801 5956 5247  2.4 38.85 38.13 45.82 42.22 47.54 1165 1030 1230  43.99 60.9 1785 1579  2.3 37.19 36.53 42.06 38.74 43.53 1078 1024 906  42.32 56.38 1589 1405  32.28 1.7 27.17 32,80 27.0 2 22.97 21.17 23.23 70 5 421 624 366 414 33.92 28.81 36.22 25.76 23.73 26.13 819 502 724 434 491 2806O  (g)  (5)  3.8 67.30 62.11 143.9 1801 60 . 3 8 114.7 105.7 121.4 6484 4620 4083 502 9 5677  1.8  Terms i n equation (27 (d) (b) (c) (e) (f)  750  750  300  300  149  112  145  53  T  R  1  A.  186  119  145  51  T  R  I  27  69  T  R  25  68  T  R  11  64  T  R  10  63  T  R  6  158  117 I  118 I  111 I  A 43  A 46  A 55  A  116 •  T  R  I  6  127  113  T  R  I  L  A 53  A 54  A  (1)  375 -2960  L  (2)  336 -2103  L  (1)  355 -2049  L  (2)  557 -1370  L 580  112 I  386 -2787  L 662  L 639  L  (1) -1318 (2)  -1005 . (1) -10 57 (2)  108 TABLE 3.  CALCULATION OF INELASTIC BENDING THEORY SOLUTION FOR fEiST BEAM 2. 1  - continued  Terms i n equation (27) T o t a l m by Outer (Central Load t P Defln. i n t e r p o - Ho ' Reac'-i Ii e a c - / m t i o n (k) t i o n (h) (i) (j) lation kips R (11) (7) <ips 0 2 kips 03) (14) 2  2  di  -153  -113  -195 -573  401  -190  -121 -20 9 -708  65  -29  -71  -110 -121  -41  -26  -70  -109 -113  46  -11  -65  -63  -73  -39  -10  -64  -63  -70  46  -6  -153 -49  -52  25  -6  -127 -50  -55  -56  2  69.69  1.13 4.51  28o66  33.2  59.61  1.29 3.39  26.30  30.2  39.40  1 93  21.43  24.0  19.10  21.0  27.51  0  2.53  2.5< ) 1.92  TABLE 9 .  CALCULATION OF INELASTIC BENDING THEORY DEFLECTIONS FOR TEST BEAM 2  Units: €- i n . / ^ x 10" ; 3  £o mo l[ u^l)  m  2  ^2  n  3  m n  4 4  3  mi UjfVZ  90 78.92 251.5 14350  m^ n u  8 6896  3  Terms i n E q u a t i o n (29)  m  Defl. Sn i n . (8)  1001  1655  -1381  -269 156  555  1.217  863  1013  -1003  -250 80  88  0.791  0.357  562  667  -418  -409  50  54  0.506  11 -146 0.251  426  505  -167  -472  37  40  0.369  810 7  -123 1.011  69  117  45  432  6  - 7 0 0.599  66  112  56  130 10 -67  72  (t)  (q)  51  115 * 53  (s)  (p)  (m) (n) (h)  145  27.51 27.68 32.79 23.32 24.12 146 445 437 21.95 17.50 337 273  Total  (r)  (e)  6 9 . 6 9 71.91 77.11 156.9 173.5 311.6 121 7376 3552 202.3 96.2 141.8 6346 354.4  37.40 33.10 43.23 44.01 45.70 5 3.33 1096 1160 54.21 30.24 40.57 625 969  Defl. Si_in. (5)  (d)  (c)  Terms i n E q u a t i o n (30)  Total  (3)  59.61 6132 66.49 48 111.3 113.3 140.0 76.82 4440 196.7 4839 127.5 103.0 71.5 10240 2273 3927 20 73.87 165.4 7976  m, n and u - k i p s / i n ? x 1 0 " ; d e f l e c t i o n - i n . x 10 unless stated.  (0)  IS  110  TABLE 10.  MOMENTS AND DEFLECTIONS CALCULATED FROM INELASTIC BENDING THEORY  Load Kips  §K  in.  in.  1.13  1.011  1.217  258  1.29  0.599  0.791  145  168  1.98  0.357  0.506  277  107  127  2.59  0.251  0.369  214  77  92  2.78  MLhi kipin.  MLho kipi n.  * 33.2  306  270  299  30.2  298  231  24.0  287  21.0 15.8  * Failure  Remarks  ' ^L  Msh kipin.  Mh s  M  Lho  Inelastic Theory  E l a s t i c beam theory i n c l u d i n g shear deformation  curvature  TABLE 11. PREDICTED AND TEST MOMENTS AND DEFLECTIONS AT FAILURE  Failure Load  Msh kipin.  MLhi L h o Msh kip- kipMLhi in. in.  ^L  %  in.  in.  ^1.02 >1.52  71.52  M  Test Moments and d e f l e c t i o n s measured at 33.8 k i p s  34.1  ^294  I n e l a s t i c Bending Theory  33.2  306  270  299  1.13  1.01  1.22  Theory o f L i m i t Design  31.0  232  255  282  1.11  0.63  0.88  >288 >317  112  CONCLUSIONS AND RECOMMENDATIONS Theory o f L i m i t  Design  According t o the theory o f l i m i t design; when a redundant frame s t r u c t u r e i s l o a d e d beyond t h e l i m i t o f e l a s t i c s t r a i n , r e d i s t r i b u t i o n o f moments t a k e s p l a c e u n t i l a " p l a s t i c h i n g e " mechanism i s formed and t h e s t r u c t u r e d e f l e c t s a p p r e c i a b l y under c o n s t a n t load.  When t h e t e s t beam was l o a d e d beyond t h e e l a s t i c  l i m i t , r e d i s t r i b u t i o n o f moments took p l a c e u n t i l a " p l a s t i c h i n g e " mechanism had formed - but t h e s t r u c t u r e f r a c t u r e d a f t e r some d e f o r m a t i o n .  I f the t e s t  ure had deformed c o n s i d e r a b l y w i t h o u t f r a c t u r e ,  structthen  i t would have been f a i r l y s a f e t o c o n c l u d e t h a t t h e theory o f l i m i t design adequately p r e d i c t s the f a i l u r e l o a d o f aluminum frames i n t h e same way as i t has been  2 shown t o p r e d i c t f a i l u r e l o a d s o f m i l d s t e e l  frames.  Thus, a l t h o u g h t h e t h e o r y o f l i m i t d e s i g n p r e d i c t e d t h e f a i l u r e l o a d o f t h e t e s t beam, t h e t y p e o f f a i l u r e  which  took p l a c e i n d i c a t e s t h a t o t h e r s t r u c t u r a l c o n f i g u r a t i o n s w i l l f a i l b e f o r e t h e l i m i t d e s i g n c o l l a p s e mechanism i s realized.  F o r i n s t a n c e i t may n o t be r e l i a b l e t o assume  t h a t s e t t l e m e n t o f s u p p o r t s does not a f f e c t t h e f a i l u r e l o a d o f a c o n t i n u o u s beam.  113-  A n o t h e r c o n c l u s i o n from t h e beam t e s t s w i t h regard  to t h e theory o f l i m i t design  shear f o r c e i s l a r g e i . e . when V/  i s t h a t when the approaches t h e  w y i e l d s t r e s s f o r pure s h e a r t h e p l a s t i c moment capa c i t y o f t h e beam s e c t i o n i s reduced. Theory o f I n e l a s t i c Bending According  t o t h e i n e l a s t i c bending  theory,  f a i l u r e i n a redundant frame s t r u c t u r e o c c u r s a t t h e t e n s i o n s i d e o f t h e c r o s s s e c t i o n when t h e moment as determined by t h e s i m p l e a maximum v a l u e .  s t r e s s - s t r a i n curve reaches  The t e s t beam f a i l e d by f r a c t u r e o f  the t e n s i o n f l a n g e i n accordance w i t h t h e i n e l a s t i c theory.  However, due p a r t l y t o h i g h shear f o r c e  ure c r a c k s a l s o o c c u r r e d  fail-  a d j a c e n t t o one o f t h e s t i f f -  ener hol'es. The  presence o f h i g h shear f o r c e reduced  the b e n d i n g c a p a c i t y o f t h e beam between t h e l o a d  point  and t h e c e n t r a l s u p p o r t and t h e r e f o r e changed t h e d e f o r m a t i o n c o n d i t i o n o f t h e beam from t h e i n e l a s t i c bending theory.  The t h e o r y  predicts that large  inelastic  s t r a i n s o c c u r a t t h e o u t e r s t i f f e n e r h o l e s under under the l o a d p o i n t , v/hereas d u r i n g t h e t e s t , l a r g e  inelastic  11U  s t r a i n s f i r s t o c c u r r e d a t t h e i n n e r h o l e s where t h e s h e a r f o r c e was l a r g e .  Because o f t h e decrease  i n bend-  i n g c a p a c i t y due t o h i g h shear f o r c e , t h e r e was a g r e a t e r moment r e d i s t r i b u t i o n t h a n p r e d i c t e d by t h e t h e o r y ( Mh o f 1.02 compared t o 1.13 from t h e i n e l a s t i c MLhi t h e o r y and 1.11 from l i m i t d e s i g n t h e o r y ) .  bending  S  When t h e t e s t beam f a i l e d , t h e d e f l e c t i o n under t h e l o a d p o i n t was more t h a n 1 . 5 2 i n c h e s compared t o a p r e d i c t e d d e f l e c t i o n a t f a i l u r e o f 1.01 i n c h e s from the i n e l a s t i c bending t h e o r y , i n d i c a t i n g t h a t t h e c u r v a t u r e over t h e c e n t r a l support when f a i l u r e o c c u r r e d was g r e a t e r t h a n p r e d i c t e d from t h e t h e o r y . One o f t h e purposes o f t h e beam t e s t s was t o check t h e c o r r e c t n e s s o f t h e moments and d e f l e c t i o n s p r e d i c t e d by t h e i n e l a s t i c bending t h e o r y beyond t h e elastic limit.  Prom t h e e l a s t i c l o a d l i m i t  (17 k i p s )  t o about 27 k i p s , t h e moments p r e d i c t e d by t h e i n e l a s t i c t h e o r y were w i t h i n c a l c u l a t i o n e r r o r o f t h e t e s t moments ( F i g . 2 2 ) . Beyond a l o a d o f 27 k i p s t h e s e c t i o n o f t h e beam a t t h e h o l e s began t o y i e l d i n shear and t h e mome n t s became d i s t r i b u t e d i n t h e beam d i f f e r e n t l y  than  p r e d i c t e d by t h e i n e l a s t i c bending t h e o r y ( F i g . 2 2 ) . The d e f l e c t i o n c u r v e s from t h e t h e o r y f o l l o w e d t h e same shape as t h e t e s t d e f l e c t i o n c u r v e s but t h e t h e o r e t i c a l  deflect-  115  i o n s were 10 t o 20 p e r cent l e s s than t h e t e s t  deflect-  i o n s ( F i g . 25). One r e a s o n f o r t h i s i s t h a t t h e t h e o r y does n o t i n c l u d e d e f o r m a t i o n due t o shear Shortcomings  strains.  o f the T e s t s (1)  When t h e t e s t arrangement was d e c i d e d  upon, i t was n o t a n t i c i p a t e d t h a t t h e beam s t i f f e n e r s would a p p r e c i a b l y a l t e r the amount o f moment b u t i o n b e f o r e f a i l u r e i n t h e t e s t beam.  redistri-  I f there are  no s t i f f e n e r s , t h e i n e l a s t i c bending t h e o r y p r e d i c t s t h a t f a i l u r e t a k e s p l a c e over t h e c e n t r a l support i n e l a s t i c "bending b e g i n s under t h e l o a d p o i n t s .  before However,  because o f t h e s t i f f e n e r s arid s t i f f e n e r h o l e s , t h e t h e o r y p r e d i c t s t h a t i n e l a s t i c bending has t a k e n p l a c e under the l o a d p o i n t s when f a i l u r e t a k e s p l a c e over t h e c e n t r a l support.  The t e s t s a r e t h e r e f o r e n o t c o n c l u s i v e i n show-  i n g whether o r not t h e t h e o r y o f l i m i t d e s i g n c a n be used t o f i n d f a i l u r e l o a d s o f frameworks made o f such non s t r a i n - h a r d e n i n g m a t e r i a l s as aluminum a l l o y (?$5S-T6). (2)  The presence  o f t h e s t i f f e n e r h o l e s comp-  l i c a t e s i n e l a s t i c bending t h e o r y a n a l y s i s o f t h e t e s t beam and s i n c e s i m p l i f y i n g assumptions  had t o be made  r e g a r d i n g t h e moment c a r r i e d by t h e s t i f f e n e r s , t h e beam r e s u l t s cannot be compared w i t h c e r t a i n t y t o t h e i n e l a s t i c bending t h e o r y .  A l s o t h e presence  o f h i g h shear f o r c e i s  116  not p r e s e n t l y t a k e n i n t o account by t h e i n e l a s t i c bending theory  and t h i s a l s o makes comparison u n c e r t a i n . (3)  indeterminate,  Because t h e beam t e s t was s t a t i c a l l y t h e moments were measured i n d i r e c t l y by  means o f s t r a i n gauges on t h e f l a n g e s .  The moments  measured were found t o be s t a t i c a l l y i n e r r o r w i t h t h e load.  A l t h o u g h s t e p s were t a k e n t o r e c t i f y t h e moments  t h e i r correctness  a t t h e approach o f f a i l u r e i s not c e r t -  ain.  Recommendations' I t i s d e s i r a b l e to t e s t the p r e d i c t i o n s o f the i n e l a s t i c bending t h e o r y and i t i s recommended t h a t t h i s can be done most e a s i l y by means o f t h e s i m p l e two-support beam t e s t under p o i n t l o a d .  The  correctness  o f t h e i n e l a s t i c t h e o r y up t o and i n c l u d i n g f a i l u r e can be t e s t e d i n a simple beam arrangement by comparing t h e t e s t moments and d e f l e c t i o n s under t h e l o a d v/ith t h e p r e d i c t e d ones. and  I f the d e f o r m a t i o n s and moments up t o  i n c l u d i n g f a i l u r e a r e p r e d i c t e d by t h e i n e l a s t i c  theory,  then the theory should  p r e d i c t the f a i l u r e def-  o r m a t i o n and l o a d i n a redundant frame p r o v i d e d f o r c e and a x i a l f o r c e a r e i n s i g n i f i c a n t .  shear  11"7 B e s i d e s i n v e s t i g a t i n g the i n e l a s t i c  bending  t h e o r y , i t i s recommended t h a t a t e s t d e m o n s t r a t i o n  be  made o f a s t a t i c a l l y i n d e t e r m i n a t e s t r u c t u r e i n w h i c h the mechanism c o n d i t i o n of l i m i t d e s i g n i s not before f a i l u r e occurs.  reached  I n the t e s t arrangement i t i s  d e s i r a b l e t h a t shear f o r c e i s not t o o g r e a t and t h a t t h e r e i s no need f o r s t i f f e n e r s which c o m p l i c a t e i n elastic analysis.  To f u l f i l l  c o n t i n u o u s beam, i t may  these c o n d i t i o n s i n a  be n e c e s s a r y  o f s u p p o r t s i n the t e s t arrangement.  to use  settlement  11$  REFERENCES  Baker, Home and Heyman. S t e e l Skeleton. Volume I I . Cambridge U n i v e r s i t y Press, 1956. H r e n n i k o f f , A. I n e l a s t i c Bending with Reference to L i m i t Design. T r a n s a c t i o n s A.3.C.E. Vol.113* 1948. Timoshenko, S. Strength o f M a t e r i a l s : P a r t I : Elementary Theory and Problems. D. Van Nostrand. 1940.  119.  NOMENCLATURE  P  -  load  L  -  l e n g t h o f t h e t e s t beam from t h e c e n t r e s u p p o r t to the outer support  V  -  shear f o r c e i n beam  R  -  r e a c t i o n o f t h e o u t e r support o f t h e t e s t beam  E  -  modulus o f e l a s t i c i t y  G  -  shear modulus o f e l a s t i c i t y  /x  -  Poisson's Ratio  = l a t e r a l ! strain fora uniaxially longitudinal strain  l o a d e d element h, t K  w  , t f , A , A f , - beam s e c t i o n p r o p e r t i e s i n T a b l e 2. w  -  r a t i o Af /  explained  x  I aA  w  I  -  moment o f i n e r t i a o f beam s e c t i o n  Z  -  e l a s t i c s e c t i o n modulus o f beam s e c t i o n  -  p l a s t i c s e c t i o n modulus o f beam s e c t i o n  Mp  -  p l a s t i c moment o f beam s e c t i o n  0~  -  y i e l d s t r e s s ( f o r A l 65S-T6, s t r e s s a t 0.2 p e r cent o f f s e t )  ^  -  s t r e s s ( f o r i n e l a s t i c bending t h e o r y - s t r e s s i n the flange)  £  -  s t r a i n ( f o r i n e l a s t i c bending t h e o r y - s t r a i n i n the flange)  m  -  u n i t f u n c t i o n f o r moment  Z  P  v  y  (Eq.(2)0  -120  4>  -  angle change  n  -  u n i t f u n c t i o n f o r angle change (eq.(4))  x  -  l e n g t h o f beam under constant shear f o r c e  S  -  deflection  U  -  u n i t f u n c t i o n f o r d e f l e c t i o n (Eq.  -  shear s t r e s s  -  u n i t f u n c t i o n f o r shear (Eq. (9))  -  u n i t f u n c t i o n f o r shear s t r e s s a t t h e c e n t r o i d  -  moment over the c e n t r a l support  -  moment a t t h e s t i f f e n e r h o l e s near the c e n t r a l support ( F i g . 22).  -  moment under the l o a d  -  moment a t the i n n e r h o l e s under the l o a d ( F i b . 22).  point  -  moment a t the outer h o l e s under the l o a d ( F i g . 22).  point  ^  -  d e f l e c t i o n under the l o a d  §k  -  d e f l e c t i o n a t 0.45L from the centre  q ^/A™  (Eq.(10))  d m  M  . L  h  l  M u~ T L  4  h  0  (6))  point  point support  

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